NEAR-BED HYDRODYNAMICS AND SEDIMENT TRANSPORT
IN THE SWASH ZONE
by
Thijs Lanckriet
A dissertation submitted to the Faculty of the University of Delaware in partial
fulfillment of the requirements for the degree of Doctor of Philosophy in Civil
Engineering
Summer 2014
© 2014 Thijs Lanckriet
All Rights Reserved
NEAR-BED HYDRODYNAMICS AND SEDIMENT TRANSPORT
IN THE SWASH ZONE
by
Thijs Lanckriet
Approved:
__________________________________________________________
Harry W. Shenton III, Ph.D.
Chair of the Department of Civil and Environmental Engineering
Approved:
__________________________________________________________
Babatunde A. Ogunnaike, Ph.D.
Dean of the College of Engineering
Approved:
__________________________________________________________
James G. Richards, Ph.D.
Vice Provost for Graduate and Professional Education
I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a
dissertation for the degree of Doctor of Philosophy.
Signed:
__________________________________________________________
Jack A. Puleo, Ph.D.
Professor in charge of dissertation
I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a
dissertation for the degree of Doctor of Philosophy.
Signed:
__________________________________________________________
Tian-Jian Hsu, Ph.D.
Member of dissertation committee
I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a
dissertation for the degree of Doctor of Philosophy.
Signed:
__________________________________________________________
James T. Kirby, Ph.D.
Member of dissertation committee
I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a
dissertation for the degree of Doctor of Philosophy.
Signed:
__________________________________________________________
Gerd Masselink, Ph.D.
Member of dissertation committee
"I do not know what I may appear to the world, but to myself I seem to
have been only like a boy playing on the sea-shore, and diverting
myself in now and then finding a smoother pebble or a prettier shell
than ordinary, whilst the great ocean of truth lay all undiscovered
before me."
Sir Isaac Newton
iv
ACKNOWLEDGMENTS
I wish to thank my advisor, Jack Puleo, for the invaluable guidance that he has
given me during my PhD. I am immensely grateful for the many enlightening
discussions, the advice, and all the opportunities he has given me to develop myself
through field work, travel, and teaching. I also treasure the countless great memories,
the many sand-covered, sleep-deprived laughs we had during field work, and all the
Duvels we imbibed. Jack truly went above and beyond as an advisor and made these
past five years an experience I will never forget.
I also want to thank my doctoral committee members, Tom Hsu, Jim Kirby,
and Gerd Masselink for their input in my research. A special thanks goes out to Gerd
Masselink for graciously hosting me as a visiting scholar at the Coastal Processes
Research Group at Plymouth University. Many thanks also go to the entire team of
the BeST field experiment: Chris Blenkinsopp, Daniel Buscombe, Daniel Conley,
Claire Earlie, Peter Ganderton, Gerd Masselink, Robert McCall, Luis Melo De
Almeida, Tim Poate, Barbara Proença, Jack Puleo, Paul Russell, Megan Sheridan, and
Ian Turner, it was an honor to count sand grains with all of you, and I hope you enjoy
reading this dissertation as a product of our work. The development of the
Conductivity Concentration Profiler (CCP) was also essential to obtaining the
measurements presented in this work. I gratefully acknowledge Michael Davidson,
Joe Faries, Betsy Hicks and Jack Puleo for developing and testing the earlier versions
of this sensor, thus paving the way for the development of the current version CCP
v
that I used. Nick Waite was the mastermind behind the development of the electronics
of the current CCP, and I am greatly indebted to Nick for his work.
Lastly, many thanks go to all my fellow students, researchers and staff at the
Ocean Engineering Lab at the University of Delaware and at Plymouth University for
the great work experience, to all my friends who supported me throughout this
journey, and to my family in Belgium, who had to miss me for the past five years.
vi
TABLE OF CONTENTS
LIST OF TABLES ......................................................................................................... x
LIST OF FIGURES ....................................................................................................... xi
LIST OF SYMBOLS AND ABBREVIATIONS ........................................................ xvi
ABSTRACT ................................................................................................................ xix
Chapter
1
INTRODUCTION .............................................................................................. 1
1.1
1.2
2
The Swash Zone ........................................................................................ 2
Sheet Flow ................................................................................................. 6
THE CONDUCTIVITY CONCENTRATION PROFILER ............................ 12
2.1
2.2
2.3
Introduction ............................................................................................. 12
Sensor Design .......................................................................................... 15
Sensor Calibration and Measurement Characteristics ............................. 20
2.3.1
Calibration ................................................................................... 20
2.3.1.1
2.3.1.2
2.3.2
Measurement Volume Quantification ......................................... 25
2.3.2.1
2.3.2.2
2.3.2.3
2.4
Theory........................................................................... 25
Lateral Extent of Current Field..................................... 28
Vertical Extent – Profile Smoothing ............................ 30
Discussion................................................................................................ 35
2.4.1
2.4.2
2.5
Existing Relationships between Concentration and
Conductivity ................................................................. 20
Experiments for CCP Calibration ................................. 22
Sensor Design .............................................................................. 35
Sensor Calibration ....................................................................... 38
Conclusions ............................................................................................. 42
vii
3
THE BEACH SAND TRANSPORT FIELD STUDY: EXPERIMENT
CONDITIONS .................................................................................................. 43
3.1
3.2
4
Field Site .................................................................................................. 43
Instrumentation ........................................................................................ 48
SHEET FLOW SEDIMENT TRANSPORT DURING QUASI-STEADY
BACKWASH ................................................................................................... 55
4.1
4.2
4.3
Measurements Quality Control................................................................ 55
Example Time Series ............................................................................... 58
Concentration Profile............................................................................... 63
4.3.1
4.3.2
4.4
4.5
4.6
5
Top and Bottom Boundary of the Sheet Layer ............................ 63
Ensemble-averaged Sediment Concentration Profiles ................ 67
Sheet Thickness and Sheet Load ............................................................. 74
Discussion................................................................................................ 79
Conclusions ............................................................................................. 83
TURBULENCE DISSIPATION ...................................................................... 85
5.1
5.2
Introduction ............................................................................................. 85
Methodology............................................................................................ 88
5.2.1
5.2.2
5.2.3
5.3
Results ................................................................................................... 100
5.3.1
5.3.2
5.3.3
5.3.4
5.4
Time Series Excerpt .................................................................. 100
Dissipation Decay Rates ............................................................ 104
Vertical Dissipation Rate Profiles ............................................. 107
Wave Height and Water Depth Dependence ............................. 109
Discussion.............................................................................................. 112
5.4.1
5.4.2
5.4.3
5.4.4
5.5
Existing Analysis Methods .......................................................... 88
Laboratory Validation ................................................................. 90
Field Experiment and Data Processing........................................ 95
Analysis Methods ...................................................................... 112
Dissipation Rate Magnitude ...................................................... 115
Vertical Dissipation Profile ....................................................... 115
Turbulence Damping by Density Stratification ......................... 116
Conclusions ........................................................................................... 120
viii
6
RECOMMENDATIONS FOR FURTHER RESEARCH.............................. 122
6.1
6.2
6.3
6.4
7
The Conductivity Concentration Profiler .............................................. 122
Sheet Flow and Swash-zone Morphology ............................................. 123
Turbulence Dissipation Rates ................................................................ 126
Negative Results .................................................................................... 127
CONCLUSIONS ............................................................................................ 129
REFERENCES ........................................................................................................... 132
Appendix
A
COPYRIGHT NOTES ................................................................................... 152
ix
LIST OF TABLES
Table 2.1 : Results of calibration experiments. ......................................................... 24
Table 2.2 : Grid spacings in the multigrid approach ................................................. 28
Table 3.1 : Offshore forcing conditions during the BeST field study. ...................... 44
Table 4.1 : Summary of fitting results for ensemble-averaged profiles. ................... 71
Table 5.1 : Surf and swash zone dissipation estimates. ............................................. 86
Table 5.2 : Turbulence dissipation rate estimates for stationary flow laboratory
test. .......................................................................................................... 91
x
LIST OF FIGURES
Figure 1.1 : Location of the swash zone on the beach. ................................................ 2
Figure 2.1 : The CCP sensor. Inset: the 32 gold-plated electrodes used for the
conductivity measurements. .................................................................... 17
Figure 2.2 : Schematic of the CCP measurement procedure. ..................................... 18
Figure 2.3 : Circuit diagram of the CCP analog conductance measurement unit. ...... 19
Figure 2.4 : Conductivity calibration for fine sand S1 (a) and coarse sand S2 (b).
Vertical bars represent 1 standard deviation on either side of the mean
of the 1900 (a) or 380 (b) samples used for each of the representative
conductivity measurements. The gray and black dashed lines are
least squares fits using the linear law (2.4) and power law (2.6)
respectively. ............................................................................................. 24
Figure 2.5 : (a) Location of the x-y plane where the normalized current field is
computed. (b) Normalized current field through x-y plane indicating
lateral extent of the measurement volume. The black curve in (b)
indicates where the current field is 1 % of its maximum value. ............. 29
Figure 2.6 : (a) Conductivity profile for a 4.0 mm prescribed conductivity
transition. Dashed line is the prescribed conductivity profile of the
medium; triangles indicate the 5 % and 95 % cutoff for the prescribed
profile. The solid line is the simulated conductivity profile for the
CCP; squares indicate the 5 % and 95 % cutoff for the simulated
profile. (b) Comparison of simulated CCP results (solid black line)
with experimental CCP measurements (gray dots) across a sand-water
interface. .................................................................................................. 31
Figure 2.7 : Sheet thickness as measured by a simulated CCP sensor compared to
sheet flow layer thickness determined without sensor smoothing.
Black line with triangles: sheet thickness determined using curvefitting technique (Section 4.3.1). Red line with circles: sheet thickness
determined via 5%-95% cutoff of conductivity profile. Blue line with
squares: sheet thickness determined via 5%-95% cutoff of
conductivity profile for miniaturized CCP sensor (Section 2.4.1).
Black solid line is line of perfect agreement. .......................................... 34
xi
Figure 3.1 : Location of the BeST field site in Perranporth........................................ 44
Figure 3.2 : Offshore wave conditions for the BeST study. a) tide level. b)
significant wave height. c) spectral peak period. d) wave direction.
Horizontal dashed line is shore normal incidence. Gray shading in all
plots indicates the sampling period. ........................................................ 45
Figure 3.3 : a) Beach profiles measured prior to high Tide 7 on October 13, 2011,
along the central, north, and south transects: the horizontal dotted line
identifies the mean high water level, the solid gray horizontal line
indicates the cross-shore extent of the scaffold rig, and the black
square the cross-shore location of the main instrument bar. b) median
grain size within the extent of the scaffold rig as a function of crossshore distance. ......................................................................................... 46
Figure 3.4 : Morphological variability during the BEST field study. a) Cumulative
elevation change for each tide: horizontal black lines denote seaward
and landward edges of the scaffold frame and horizontal white line
indicates the location of the main instrument bar. b) Cumulative
volumetric flux required to cause measured elevation changes
determined from pre- and post-tide beach profile surveys. ..................... 48
Figure 3.5 : Images showing the (a) scaffold frame and (b) main instrument bar:
sensors on the main instrument bar are (a) electromagnetic current
meters; (b) Vectrino I velocimeters; (c) Vectrino II acoustic Doppler
current profilers; (d) FOBS; (e) CCPs; the buried PTs are only
identified by their name in the image. ..................................................... 49
Figure 3.6 : Sideways view of three buried CCPs, indicated by arrows. Downhanging probes are Vectrino-II velocimeters. Yellow tape measure
indicates scale. ......................................................................................... 51
Figure 3.7 : Overview of the BeST field location displaying camera tower (A),
data logging cabins (B), instrumentation scaffolding (C). ...................... 53
Figure 4.1 : Time series excerpt of swash-zone measurements. (a) Water depth
time series. (b) cross-shore (u, solid line) and alongshore (v, dashed
line) velocity measured 0.03 m above the bed. (c) Time series of
measured concentrations at 6 elevations from CCP A. (d) Contour
time series of sediment concentrations from CCP B. Black line
indicates bottom of the sheet flow layer according to the curve-fitting
method described in section 4.3.1, blue line indicates bottom of sheet
flow layer according to concentration cut-off (c = 0.51, see section
4.3.1), magenta line indicates top of sheet flow layer (c = 0.08). ........... 59
xii
Figure 4.2 : Time series excerpt of swash-zone measurements. (a) Water depth.
(b) Cross-shore (solid line) and alongshore (dotted line) velocity.
Thick black line indicates when quasi-steady backwash sheet flow
occurred that complied with all quality-control criteria. (c) Sediment
volume fraction measured by CCP. Magenta (black) line indicates top
(bottom) of sheet layer, yellow vertical lines indicate times when
instantaneous profiles are displayed in panels d-g. (d-g) Instantaneous
sediment volume fraction profiles measured by two collocated CCPs
(CCP A and CCP B) during four instances of the event, indicated by
yellow lines in panel c. ............................................................................ 61
Figure 4.3 : Instantaneous sediment concentration profiles recorded during quasisteady backwash events at (a) 17:30:15.75 UTC (Figure 4.2d) and (b)
17:32:1.50 UTC (Figure 4.2g). Black line indicates the measured
concentration profile. Blue line indicates curve fit of the form of
equation 4.3. Red diamond indicates inflection point in curve. Red
upward facing triangle and red downward-facing triangle indicate the
top and bottom of the sheet layer, 𝒛𝒕 and 𝒛𝒆, respectively. .................... 65
Figure 4.4 : Ensemble-averaged sediment concentration profiles for sheet
thicknesses of 6 mm (left column), 11 mm (middle column) and 16
mm (right column). Gray lines indicate individual profiles; dotted
black line indicates ensemble-averaged profile. Horizontal lines
indicate top and bottom of sheet layer. Top row: Orange solid line
indicates linear fit. Middle row: blue solid line and orange dashed line
indicate power law and linear law components of the composite curve
fit. Bottom row: orange solid line indicates ODW curve with
𝛼 = 1.73 fixed......................................................................................... 68
Figure 4.5 : Individual sheet flow sediment concentration profiles normalized by
sheet thickness. Black line indicates ensemble average of all
normalized profiles, white dotted line indicates ODW curve with
𝛼 = 1.73. ................................................................................................. 73
Figure 4.6 : (a) Sheet flow thickness 𝛿𝑠 as a function of mobility number 𝜓, (b)
Sheet load 𝐶 as a function of mobility number 𝜓. Solid line indicates
best fit through origin; dashed lines indicate factor two range. No
measurements are displayed with sheet layer thicknesses less than 5
mm as they cannot be resolved by the CCP. ........................................... 75
xiii
Figure 4.7 : Time series excerpt of swash zone measurements. (a) Water depth.
(b) Cross-shore (solid line) and alongshore (dashed line) velocity. (c)
Mobility number. Thick black lines in panels b and c indicate times
when quasi-steady backwash sheet flow occurred. (d) Sheet thickness
and (e) sheet load. Black dots indicate measurements during quasisteady backwash, gray dots indicate measurements during uprush
sheet flow events that met quality control criteria and during
backwash sheet flow events when no current meter data was available.
Orange crosses indicate predictions by fits through origin (equations
4.18-19). .................................................................................................. 77
Figure 5.1 : a) Structure function D for vertical velocity estimates 𝑤1(triangles)
and 𝑤2 (circles). Solid lines indicate fits of equation (7). b)
Wavenumber spectrum averaged over 24-minute record. c) Temporal
frequency spectrum. Error bar indicates 95% confidence band of
spectral estimate. ..................................................................................... 93
Figure 5.2 : Main instrument rig with three profiling velocimeters indicated by
white arrows. ........................................................................................... 96
Figure 5.3 : Example image of the video collection, including location of pixel
intensity time series points and timestack transect. Image brightness
and contrast was enhanced compared to raw image for increased
visibility. .................................................................................................. 99
Figure 5.4 : Time series excerpt of measurements taken during tide 5 at 16:33:10 –
16:35:10 UTC on 12 October 2011. a) Free surface elevation 𝜂. b)
Cross-shore velocity at z = 0.02 m. c) Turbulence dissipation
estimates averaged across the vertical for lower (blue line), middle
(red line) upper (green line) velocimeter. d) Coefficient of
determination 𝑅2 of equation (5.7) fit. e) Base-10 logarithm of
turbulence dissipation 𝜖. Black line in panels d and e indicates bed
level estimate from velocimeter bed level scan. f) Normalized pixel
intensity from video camera. Thick black lines indicate high pixel
intensity events. ..................................................................................... 101
Figure 5.5 : Pixel intensity timestack. Black line indicates cross-shore position of
the velocimeters. .................................................................................... 103
xiv
Figure 5.6 : Time series excerpt of turbulence dissipation decay rates, recorded
during tide 8 at 06:28:56 – 06:31:58 UTC on 14 October 2011. a)
Free surface elevation 𝜂. Dots indicate start times 𝑡0 for three boredissipation events. b) Cross-shore velocity measured at 0.02 m above
the bed. c) Measured vertically-averaged dissipation rate 𝜖 and fitted
decay rates according to equation (5.9). 𝑅2 goodness-of-fit was 0.92,
0.87 and 0.94 for the three events, respectively. ................................... 105
Figure 5.7 : Phase space averaged vertical dissipation profiles (black solid line)
and 25% and 75% percentiles of averaged profiles (gray dotted lines).
Axes labels and scales are identical for all panels. Downslope
gravitational acceleration corresponds to 𝑑𝑢∞(𝑡)𝑑𝑡 = −0.22 m/s2.
N-values indicate average number of dissipation rates used in the
vertical profile. Panel indicated by bold frame was compared against
log-layer scaling (Section 5.4.4)............................................................ 108
Figure 5.8 : a) Turbulence dissipation as a function of local water depth. b)
Inundation percentage. Error bars show variability (1 standard
deviation) between results from 10 tide measurement cycles. .............. 110
Figure 5.9 : Distributions of turbulence dissipation rates binned according to local
water depth. Different curves display results from the 10 tide
measurement cycles. .............................................................................. 111
Figure 5.10 : Repeat of time series excerpt displayed in Figure 5.4. Time series
excerpt of measurements taken during tide 5 at 16:33:10 – 16:35:10
UTC on 12 October 2011. a) Free surface elevation 𝜂. b) Cross-shore
velocity at z = 0.02 m. c-e) Turbulence dissipation estimates
calculated using an averaging window of 2.5 s (c), 1.5 s (d) and 3.5 s
(e) averaged across the vertical for lower (blue line), middle (red line)
upper (green line) velocimeter. ............................................................. 113
Figure 5.11 : Phase-space averaged velocity profile for bin −1.25 m/s ≤ 𝑢∞ ≤
−0.75 m/s, −0.50 m/s2 ≤ 𝑑𝑢∞𝑑𝑡 ≤ 0 m/s2. Gray dotted lines
indicate 25% and 75% percentiles of averaged velocity profiles. ......... 119
xv
LIST OF SYMBOLS AND ABBREVIATIONS
ADCP
Acoustic Doppler Current Profiler
Bardex II
Barrier Dynamics Experiment II
BeST
Beach Sand Transport study
c
Sediment concentration (expressed as a volume fraction)
C
Sheet load
CCP
Conductivity Concentration Profiler
CCM
Conductivity Concentration Meter
d
(Generic) grain diameter
d50
Median grain diameter
D(z,r,t)
Structure function
UDM
Ultrasonic Distance Meter
E (k)
Wavenumber spectrum
EMCM
Electromagnetic Current Meter
f
Frequency
F
Formation factor
FOBS
Fiber Optic Backscatter Sensor
g
Gravitational acceleration
h
Water depth
Hs
Significant wave height
I
Normalized pixel intensity
IBC
In-Situ Bed Camera
xvi
k
Wavenumber
m
Calibration coefficient for Archie’s law
NTP
Network Time Protocol
O(…)
On the order of magnitude of
OBS
Optical Backscatter Sensor
P(f)
Frequency spectrum
𝒫
Turbulence production rate
PIV
Particle Image Velocimetry
PV
Profiling Velocimeter
r
Pearson correlation coefficient, separation distance
R2
Coefficient of determination
RBG
Red-Blue-Green (visible-band)
𝑅𝑖
Gradient Richardson number
s
relative density of sediment
S
Turbulence reduction due to stratification
Tp
Spectral peak period
u
Cross-shore velocity
𝑢∗
Friction velocity
UTC
Coordinated Universal Time
v
Alongshore velocity
w
Vertical velocity
𝛽
Bottom slope
𝛿𝑠
𝜖
Sheet flow layer thickness
Turbulence dissipation rate
xvii
𝜖̅
Vertically averaged dissipation rate
𝜃
Shields number
𝜈
Molecular kinematic viscosity
𝜋
Dessert
𝜌𝑠
Sediment density
𝜅
Von Kármán constant
𝜈𝑡
Eddy viscosity
𝜌𝑓
Density of water (in the absence of sediment)
𝜎𝑐
Prandtl-Schmidt number
𝜎𝑚
Electrical conductivity of sand/fluid mixture
𝜎𝑓
Electrical conductivity of fluid in absence of sediment
𝜏𝑏
Bed shear stress
Φ
Volumetric sheet load
𝜙
Porosity
𝜓
Mobility number
xviii
ABSTRACT
The swash zone, the section of the beach that is alternatingly inundated and
exposed as a result of wave runup, is an important part of the nearshore coastal zone
because of hydrodynamic and sediment transport processes that affect the morphology
and ecology of the broader littoral area. The understanding of these swash-zone
processes is poor and suffers from a lack of detailed measurements under natural
conditions. A new comprehensive dataset of swash zone processes was collected
during the Beach Sand Transport (BeST) field study, which was conducted in
Perranporth (UK) in October 2012. Measurements were taken for approximately 3
hours around high tide during 10 consecutive tidal cycles, and included several
innovative measurement techniques. First, a novel Conductivity Concentration
Profiler (CCP) was developed that measures sediment concentration in the sheet flow
layer using electrical conductivity as a proxy. The CCP measures a 29-point
conductivity profile at 1 mm resolution by multiplexing through a vertical array of 32
plate electrodes. The relationship between conductivity and sediment concentration
was calibrated during lab experiments with known masses of sediment neutrally
suspended in a heavy liquid. The horizontal and vertical extent of the CCP
measurement volume was analyzed using a numerical model of the electric field
around the sensor, and indicated that sheet flow layers with a thickness greater than 5
mm are resolved accurately. CCP measurements during the BeST field study
demonstrated that sheet flow occurs frequently in the swash zone during both the
uprush and the backwash. A detailed analysis of sheet flow focusing on quasi-steady
xix
backwash events (when effects of phase lags, surface-generated turbulence and
accelerations were small) showed that the sheet flow sediment concentration profile
has a linear shape in the lower section of the sheet flow layer and a power-law shape
in the upper section. The shape of the concentration profile is self-similar and can be
described by a single curve for sheet flow layer thicknesses ranging from 6 to 18 mm.
The sheet flow layer thickness and sheet load (the sediment mass mobilized in the
sheet flow layer) are well-correlated with the hydrodynamic forcing represented by the
mobility number. Secondly, high-resolution near-bed velocity profiles were measured
by three profiling acoustic Doppler velocimeters and were used to estimate near-bed
turbulence dissipation rates, derived from the structure function. Dissipation rates
between 6 ⋅ 10-5 m2/s3 and 8 ⋅ 10-3 m2/s3 were observed. Temporal phasing of strong
turbulence dissipation events agreed well with remotely sensed pixel intensity
associated with wave breaking. Bore-generated turbulence decayed rapidly following
bore arrival and followed a decay rate similar to grid turbulence. Vertical dissipation
profiles demonstrate that turbulence was dominated by (advected) bore-generated
turbulence during the uprush and initial stages of the backwash, and by bed-generated
turbulence during the later stages of the backwash. A scaling analysis shows that near
the bed, sediment-induced density stratification effects on the turbulent kinetic energy
budget may have been an order of magnitude larger than turbulence dissipation.
xx
Chapter 1
INTRODUCTION
This dissertation discusses hydrodynamic and sediment transport processes
that take place in the swash zone of sandy beaches. A First International Workshop on
Swash Zone Processes was convened in Lisbon, Portugal in September 2004 [Puleo
and Butt, 2006] to synthesize swash zone research findings and identify a number of
key needs for future research. Several recommendations made during the workshop
formed the basis for the work presented in this dissertation, including the need for
comprehensive field measurements of swash zone processes in general and
measurements of turbulence and sheet flow sediment transport in particular [Masselink
and Puleo, 2006]. A second workshop will take place in Newark, Delaware, in July
2014, ten years after the first workshop and roughly at the time of printing of this
dissertation. This workshop will provide a new synthesis of the swash zone research
conducted since the previous workshop, as well as a new set of recommendations for
future research, and bring the process full circle.
This work is organized as follows. A concise overview of the state of the art
on the swash zone and sheet flow processes is given in the remainder of this chapter.
Chapter 2 discusses the development of the Conductivity Concentration Profiler, a
new sensor to measure sediment concentration in the sheet flow layer. Experimental
conditions are described in Chapter 3 for the Beach Sand Transport (BeST) field
experiment which provided the measurements discussed in this work. Results from
the field measurements of sheet flow processes are discussed in Chapter 4 and of
1
turbulence dissipation rates in Chapter 5. Chapter 6 provides a set of
recommendations for further research, and general conclusions are given in Chapter 7.
1.1
The Swash Zone
The swash zone is defined as the section of the beach that is intermittently
submerged and subaerial as a result of wave action, i.e. the area between wave run-up
and run-down [Elfrink and Baldock, 2002]. Other definitions have also been
proposed; e.g. Puleo et al. [2000] define the seaward edge of the swash zone as the
location where bore turbulence begins to significantly affect the bed.
The swash zone is an area of active research interest for various reasons
[Bakhtyar et al., 2009a]. First, it is an important area for beach morphodynamics
since it is an area of intense sediment transport as well as a sediment transport
boundary condition. In the cross-shore, the swash zone is the place where sediment
exchange between the subaqueous and subaerial zones of the beach and hence
shoreline change occur [Masselink and Hughes, 1998]. In the alongshore sense,
between 5% and 60% of the total alongshore sediment transport takes place in the
Figure 1.1:
Location of the swash zone on the beach.
2
swash zone, depending on the breaker type [Bodge, 1989; Wang et al., 2002]. For
aeolian sediment transport on the subaerial beach, the swash zone acts as the seaward
boundary where sediment pick-up begins [Bauer and Davidson-Arnott, 2003].
The swash zone is also important for nearshore hydrodynamics. Wave runup
processes that take place in the swash zone are crucial to coastal safety and dune
erosion [Stockdon et al., 2006]. The swash zone is also the landward boundary for
breaking wave energy dissipation and turbulence generation. Breaking wavegenerated turbulence is exchanged between the surf zone and the swash zone, with the
swash zone acting as a sink for surf zone turbulence [Sou et al., 2010]. The swash
zone is also a crucial part of the beach ecosystem [Moreno et al., 2006]. Biological
processes that take place in the swash zone are affected by morphodynamic (physical)
processes. For example, the horseshoe crab (Limulus polyphemus) buries its eggs in
the top 0.20 m of the sediment bed in the swash zone, and some eggs are lost due to
exhumation by wave-induced sediment transport [Jackson et al., 2014].
Despite their recognized importance, a thorough understanding of swash zone
processes is still lacking. The main reason for this is that taking measurements in the
swash zone is difficult since swash flows are shallow, ephemeral and frequently laden
with sediment and bubbles [Puleo et al., 2014b]. Furthermore, rapid bed level
fluctuations [Puleo et al., 2014a] hinder the accurate positioning of sensors. Reviews
of swash zone processes were given by Elfrink and Baldock [2002], Longo et al.
[2002], Puleo and Masselink [2006], and Bakhtyar [2009a]. In the years since these
reviews have been published, research has continued in the form of laboratory
[Masselink et al., in preparation; Barnes et al., 2009; Sou et al., 2010; O’Donoghue et
al., 2010; Alsina and Cáceres, 2011; Sou and Yeh, 2011; Alsina et al., 2012; Astruc et
3
al., 2012; Cáceres and Alsina, 2012; Kikkert et al., 2012; Othman et al., 2014],
numerical [Calantoni et al., 2006; Hsu and Raubenheimer, 2006; Guard and Baldock,
2007; Puleo et al., 2007; Zhang and Liu, 2008; Alsina et al., 2009; Bakhtyar et al.,
2009b, 2010; Barnes and Baldock, 2010; Briganti et al., 2011; Steenhauer et al.,
2012; van Rooijen et al., 2012; Torres-Freyermuth et al., 2013; Desombre et al.,
2013] and field [Aagaard and Hughes, 2006; Baldock and Hughes, 2006; Masselink
and Russell, 2006; Turner et al., 2008; Baldock et al., 2008; Russell et al., 2009;
Masselink et al., 2009, 2010; Puleo, 2009; Blenkinsopp et al., 2010a, 2011; Austin et
al., 2011; Puleo et al., 2012, 2014b] studies.
In recognition of the fact that the swash zone poses a uniquely difficult
measurement environment, many of these recent studies have presented innovative
measurement techniques, including acoustic distance meters [Turner et al., 2008],
lidar [Blenkinsopp et al., 2010a] and stereovision [Astruc et al., 2012] to measure bed
level change and free surface elevation, a shear plate to directly measure bed shear
stress [Barnes et al., 2009], a next-generation acoustic profiling velocimeter [Puleo et
al., 2012], and a new conductivity concentration profiler for sheet flow that is
described in this work. Several laboratory studies were conducted in large wave flume
facilities [Masselink et al., in preparation; Alsina and Cáceres, 2011; Alsina et al.,
2012; Astruc et al., 2012; Cáceres and Alsina, 2012]. In smaller flumes, dam breaks
have become an increasingly popular alternative to conventional (paddle-type) wave
makers for generating swash events since dam breaks can generate larger runup events
[Barnes et al., 2009; O’Donoghue et al., 2010; Kikkert et al., 2012; Othman et al.,
2014]. Dam breaks also generate repeatable swash events that are suitable for
ensemble-averaging, e.g. to calculate turbulence characteristics [Kikkert et al., 2012].
4
An important finding of swash-zone sediment transport measurements of the
last two decades is that conventional sediment transport formulations such as the
energetics-type formulation described by Bagnold [1966a], Bowen [1980] and Bailard
[1981] or the Meyer-Peter and Müller [1948] equations, both of which relate the
sediment transport to the flow velocity cubed, do not lead to satisfactory results when
applied to the swash zone [Hughes et al., 1997; Masselink and Hughes, 1998; Puleo et
al., 2000; Nielsen, 2002; Butt et al., 2004, 2005; Masselink et al., 2005; Masselink and
Russell, 2006; Othman et al., 2014]. Additional processes that are not included in the
flow velocity cubed law significantly affect sediment processes, including sediment
mobilization by pressure gradients [Butt and Russell, 1999; Drake and Calantoni,
2001; Nielsen, 2002; Puleo et al., 2003; Baldock and Hughes, 2006; Calantoni and
Puleo, 2006; Othman et al., 2014], advected sediment and bore turbulence [Puleo et
al., 2000; Butt et al., 2004; Jackson et al., 2004; Aagaard and Hughes, 2006], swashswash interactions [Hughes and Moseley, 2007] and groundwater in- and exfiltration
[Nielsen, 1998; Turner and Masselink, 1998; Butt et al., 2001; Karambas, 2003;
Masselink and Turner, 2012; Steenhauer et al., 2012]. It is still unclear, however,
how these processes interact and ultimately determine the swash-zone sediment
budget.
More recent field studies have challenged the long-standing notion that net
sediment transport over an individual swash event is difficult to predict because net
transport is a small signal that is the difference of two large signals, the uprush and
backwash transport [Osborne and Rooker, 1999]. It is still agreed that uprush and
backwash sediment transport are large, but recent measurements have shown that net
sediment transport, and resulting bed level changes, over individual swash events are
5
not necessarily small. Event-scale bed level changes show significant variability and
events with a bed level change of the same order as the net change over a tidal cycle
frequently occur [Masselink et al., 2009; Russell et al., 2009; Blenkinsopp et al., 2011;
Puleo et al., 2014a]. It may therefore be the case that the long-term net bed level
change is driven by a slight shift in the distribution of event-by-event bed level
changes [Russell et al., 2009] or by ‘large’ swash events that are infrequent but cause
large bed level changes [Puleo et al., 2014a]. Predicting net sediment transport on the
timescale of hours or longer using small-scale, process-based models may therefore be
unsuccessful even if additional processes are included, since errors made by smallscale models will accumulate over time [Masselink et al., 2009] .
Since the small water depths of swash flows limit the capacity for suspended
load transport, the large sediment transport signals must be at least in part due to nearbed (bedload and sheet flow) sediment transport [Masselink and Puleo, 2006]. Horn
and Mason [1994] deployed sediment traps on four different beaches and found that
near-bed sediment transport was the dominant mode during the backwash phase, and
that near-bed transport was also significant, and at certain locations dominant, during
the uprush. In order to improve understanding of the swash zone sediment budget, it
is therefore necessary to gain a better knowledge of sheet flow processes in the swash
zone. Since sediment traps are only capable of measuring the time-integrated (bulk)
sediment transport, other measurement techniques are needed to investigate the timevarying near-bed transport in greater detail.
1.2
Sheet Flow
Sheet flow is a form of intense near-bed transport of non-cohesive sediment,
but the exact definition of ‘sheet flow’ is not universally agreed upon. A first
6
definition stems from measurements of sediment beds exposed to oscillatory flow in
oscillatory flow tunnel experiments. Different regimes of bedforms are encountered
as flow velocities increase. When the flow velocity exceeds a certain threshold,
bedforms are washed out and the sediment bed becomes flat, with a dense layer of
sediment moving over it [Bagnold and Taylor, 1946; Dingler and Inman, 1976]. In
this sense, sheet flow is defined as ‘near-bed sediment transport of such high intensity
that bedforms are not present’. A second characteristic of sheet flow is that
intergranular interactions are the dominant sediment mobilization mechanism (as
opposed to turbulent suspension). A second definition of sheet flow is therefore ‘a
layer of mobilized sediment where intergranular interactions are the dominant
mobilizing factor’ [Wilson, 1987; Hsu et al., 2004]. The difference in definition is
important because the criterion for the onset of sheet flow may differ depending on the
definition of sheet flow: a layer of sediment may be mobilized by intergranular
interactions before bedforms are fully washed out, a situation that constitutes sheet
flow under the second definition but not the first. Since bedforms are generally nonexistent in the swash zone, the second definition (related to intergranular collisions) is
used in this work. An interesting historical fact is that the sheet flow layer was
referred to as a ‘carpet’ before the term ‘sheet’ was used [Bagnold, 1966a]. It is noted
that the phrase sheet flow is also used in the context of rainfall runoff occurring as a
thin film of water [Julien and Simons, 1985] but that process is not related to the sheet
flow sediment transport process discussed in this work.
Sheet flow is sometimes referred to as a particular sub-type of bedload
transport and sometimes seen as a transport mode separate from bedload. In this
work, bedload and sheet flow sediment transport are regarded as two different
7
phenomena and the grouping of the two is referred to as near-bed sediment transport.
Bedload is defined here as the process of individual grains rolling, sliding or saltating
over the sediment bed without experiencing near-constant collisions with other
mobilized grains (other than the bed). The bed load layer therefore has a thickness on
the order of one grain diameter (although saltating grains might move higher above the
bed). In contrast, sheet flow consists of a layer of mobilized sediment with a thickness
of many [O(10-100)] grain diameters that are in constant collision and interaction, so
that the sheet flow layer becomes a section of the flow with its own rheology
[Armanini et al., 2005]. It is clear that there are cases where the near-bed transport is
purely bedload (e.g. a cobble-bed stream where the critical shear stress for sediment
mobilization is only occasionally exceeded) or sheet flow (e.g. sheet flow with a
thickness of over 10 mm over a sand bed, as frequently occurs in the swash zone, see
Figure 4.1) but there are also cases that are intermediate between bedload and sheet
flow.
The onset of sheet flow is typically defined using the Shields number 𝜃
[Shields, 1936]:
𝜃=𝜌
𝜏𝑏
(𝑠−1)𝑔𝑑
𝑓
(1.1)
where 𝜏𝑏 is the bed shear stress, 𝜌𝑓 is the clear-water fluid density, 𝑠 = 𝜌𝑠 /𝜌𝑓 is the
relative density of the sediment, 𝜌𝑠 is the density of the sediment, 𝑔 is gravitational
acceleration and 𝑑 is a characteristic grain diameter (typically, d50, the median grain
diameter). The bed shear stress is generally quantified using a quadratic drag law and
an empirical friction coefficient. Sheet flow is typically said to occur when θ exceeds
a threshold value of 1.0 [Nielsen, 1992] although lower threshold values have also
been proposed, particularly for smooth spherical particles [Horikawa et al., 1982;
8
Asano, 1995]. The sediment mobility number 𝜓 is also used to describe the onset of
sheet flow sediment transport:
𝑢2
∞
𝜓 = (𝑠−1)𝑔𝑑
(1.2)
where 𝑢∞ is the fluid velocity outside the boundary layer [Asano, 1995; Dingler and
Inman, 1976; Horikawa et al., 1982]. This formulation has the advantage that 𝑢∞ is
typically easier to quantify than 𝜏𝑏 and alleviates the need for an empirical friction
coefficient. Dingler and Inman [Dingler and Inman, 1976] state that sheet flow occurs
for 𝜓 > 240. An estimate of the dimensionless thickness of the sheet flow layer
under stationary flow is
𝛿𝑠
𝑑
= 10𝜃
(1.3)
where 𝛿𝑠 is the sheet flow layer thickness [Wilson, 1987]. The transition in sediment
concentration from a packed bed at the bottom boundary of the sheet flow to dilute
conditions at the top boundary has previously been described as approximately linear
[Wilson, 1987; Sumer et al., 1996; Pugh and Wilson, 1999].
Sheet flow is frequently observed in coastal settings, e.g. on the crests of
sandbars [Nielsen, 1992; Hassan and Ribberink, 2005] and in the swash zone [Hughes
et al., 1997; Masselink and Puleo, 2006]. Sheet flow has also reportedly been
observed in rivers during flood stage [Wang and Yu, 2007]. However, care must again
be taken regarding the terminology. For example, it is uncertain whether the ‘bed load
sheets’ (maximum Shields number θ = 0.3) reported by Dinehart et al. [1992],
constituted sheet flow. Sediment concentrations [O(1 kg/l)] in the sheet flow layer
exceed maximum concentrations suspended higher in the water column [O(0.1 kg/l)]
[Beach and Sternberg, 1988, 1991; Puleo et al., 2000; Butt et al., 2004; Masselink et
al., 2005]) by an order of magnitude, resulting in large sediment transport rates.
9
The first measurements of unidirectional sheet flow were conducted in annular,
parallel-plate shear cells and provided constitutive relationships for the grain stresses
that are the mobilizing mechanism for sheet flow [Bagnold, 1954; Savage and
Mckeown, 1983; Hanes and Inman, 1985]. Experiments in recirculating flow tunnels
[Shook et al., 1982; Sumer et al., 1996; Pugh and Wilson, 1999] provided additional
insights into the velocity and sediment concentration profiles in the sheet flow layer
and the increased bed roughness due to sheet flow. Sheet flow measurements under
sinusoidal, asymmetric and irregular forcing were conducted in oscillatory flow
tunnels [Horikawa et al., 1982; Ribberink and Al-Salem, 1995; Dibajnia and
Watanabe, 1998; Dohmen-Janssen et al., 2001; Ahmed and Sato, 2003; O’Donoghue
and Wright, 2004b; Ribberink et al., 2008; van der A et al., 2010; Capart and
Fraccarollo, 2011; Ruessink et al., 2011; Dong et al., 2013]. These studies have
established thresholds for the transition between the ripple regime and the sheet flow
regime, and have provided the most detailed sediment concentration and velocity
profiles in the sheet flow layer to date. Oscillatory flow tunnel studies also elucidated
the relationship between velocity and acceleration asymmetry, phase lags between the
free-stream velocity, bed shear stress and mobilized sediment, and the net sediment
flux.
Studies in large-scale wave flumes provide a more realistic reproduction of the
coastal environment and add the effects of free surface flow such as boundary layer
streaming, which alters the net transport rate [Dohmen-Janssen and Hanes, 2002,
2005; Schretlen et al., 2010]. Numerical models yield additional insights into the
sheet flow process, but require validation by physical experiments [Hsu et al., 2004;
Calantoni and Puleo, 2006; Amoudry et al., 2008; Bakhtyar et al., 2009c; Yu et al.,
10
2010; Chen et al., 2011]. Measurements of sheet flow under natural conditions have
been scarce, mainly due to the difficulty of capturing the sheet flow layer near the
sediment bed. Bakker et al. [1988] and Yu et al. [1990] provided the only known field
measurements in the swash-zone sheet flow layer before this work, in contrast with the
large number of swash-zone field studies that focused on suspended load [e.g., Butt
and Russell, 1999; Osborne and Rooker, 1999; Puleo et al., 2000; Masselink et al.,
2005; Hughes et al., 2007; Cáceres and Alsina, 2012].
To summarize, sheet flow has been a crucial unknown in the study of swashzone sediment transport. Numerical and laboratory studies have improved the
understanding of the physics of sheet flow. The missing link is to obtain
measurements of sheet flow in the swash zone under natural conditions but this
requires a custom-designed measurement solution because existing sheet flow
measurement techniques (discussed in Section 2.1) cannot be readily deployed in the
field. The design of such a custom measurement solution, the Conductivity
Concentration Profiler (CCP), is discussed in the following chapter.
11
Chapter 2
THE CONDUCTIVITY CONCENTRATION PROFILER
2.1
Introduction
The previous chapter described how sheet flow is an important sediment
transport mode in the swash zone and measurements of sheet flow under realistic
conditions are necessary to fully understand swash-zone sediment fluxes [Masselink
and Puleo, 2006]. However, no instrument has been available that is capable of
obtaining measurements of either velocity or sediment concentration in the sheet flow
layer under field or large-scale laboratory conditions in the swash zone. This chapter
describes the development of the Conductivity Concentration Profiler (CCP), which
was developed specifically to measure sediment concentrations in the swash-zone
sheet flow layer under field conditions.
Obtaining measurements of sediment concentration and/or velocity in the sheet
flow layer is difficult, especially in field conditions, due to the small thickness of the
sheet flow layer, large sediment concentrations, and the intermittent occurrence of
sheet flow. Measurements are further complicated by rapid bed level fluctuations with
amplitudes larger than the sheet flow layer thickness [Puleo et al., 2014a] and by
sharp concentration gradients where the sediment volume fraction may change by
roughly 50 % over vertical distances on the order of 10 mm in a typical sheet flow
layer. These gradients indicate the level of detail needed in sheet flow measurements
where the vertical profile of both sediment concentration and velocity must be
measured down to the stationary bed level to compute instantaneous sediment
12
transport rates. Shook et al. [1982] and Pugh and Wilson [1999] determined vertical
sediment concentration profiles in pipelines under steady, unidirectional flow using
the absorption of gamma rays and found a linear vertical concentration profile through
most of the sheet flow layer and an exponentially decaying low-concentration tail in
the upper portion of the sheet. The sheet thickness was proportional to the bed shear
stress for a wide range of Shields numbers (1 ≤ θ ≤ 15). Similarly, Montreuil and
Long [2009] measured bedload transport using CT-scanning, and found that the bed
porosity affects bedload transport quantities. Imaging techniques through a laboratory
flume side-wall were performed by Capart et al. [2002] using Voronoï diagrams and
by Spinewine et al. [2011] using the reflection of a laser sheet. Good agreement was
found between the bulk sediment transport rates as measured by weighing the solids
discharge at the flume outlet, and the transport rates obtained by vertically integrating
the measured concentration and velocity profiles. However, the agreement was poorer
for large transport rates due to wall effects in the measurement technique.
The methods described previously can only be used in scaled laboratory
settings. To the author’s knowledge, the only known method for quantifying sheet
flow sediment concentration in prototype-scale wave flumes and natural coastal
settings is to use electrical conductivity as a proxy for sediment concentration, a
technique that was first used by Horikawa et al. [1982]. More recently, a conductivity
concentration meter (CCM) developed by Ribberink and Al-Salem [1992] was used in
a number of studies [Bakker et al., 1988; Dohmen-Janssen and Hanes, 2002, 2005;
McLean et al., 2001; O’Donoghue and Wright, 2004a; Ribberink et al., 2000;
Ribberink and Al-Salem, 1995; Sánchez-Arcilla et al., 2011; Yu et al., 1990; Hassan
and Ribberink, 2005; Van der Zanden et al., 2013]. The CCM returns only a point
13
measurement, which is insufficient to instantaneously determine the vertical
concentration profile in the sheet layer. Bakker et al. [1988] and Yu et al. [1990]
partially circumvented this problem by simultaneously deploying three instruments
with a known vertical offset. O’Donoghue and Wright [2004a] conducted
experiments in an oscillatory flow tunnel with a single CCM, executing the same
experiment multiple times with the CCM positioned at different elevations in order to
re-create the vertical concentration profile. However, this approach is not possible in
field settings or in laboratory settings with irregular wave trains. Ribberink et al.
[2000] developed a remote-controlled “CCM tank” which allowed two CCM
instruments to be repositioned in the vertical with sub-millimeter accuracy, enabling a
more accurate repositioning of the sensor but experiments still had to be carried out
multiple times in order to obtain a concentration profile [McLean et al., 2001;
Dohmen-Janssen and Hanes, 2002, 2005]. The two sensors in the CCM tank were
positioned at the same vertical elevation with an offset in the streamwise direction,
allowing the estimation of granular velocity using cross-correlation of the
concentration signal [McLean et al., 2001]. The CCM tank was recently improved
with CCM probes that move up and down to follow a certain sediment concentration
value using a real-time control system [Van der Zanden et al., 2013]. Dick and Sleath
[1991, 1992] developed a different configuration of conductivity measurements by
using fifteen point electrodes mounted flush with the flume side-wall, all measuring
with reference to a large common electrode. This geometry led to a sediment
concentration profile and a smaller measurement volume compared to the fourelectrode conductivity measurement technique used by the CCM since the measured
resistance was solely determined by the volume around the small electrode as opposed
14
to the volume between the four electrodes. Also, it caused no flow interference since
no object was placed in the flow. Wall effects, on the other hand, may have
influenced concentration measurements and the concentration profile was relatively
coarse since electrodes could not be placed near each other. Zala Flores and Sleath
[1998] modified this concept by using pairs of electrodes flush with the flume sidewall spaced 10 mm apart in the horizontal to reduce radio interference. It is noted that
the previous studies, with the exception of Bakker et al. [1988] and Yu et al. [1990],
were all conducted in laboratory settings.
To achieve the goal of measuring the sediment concentration profile in the
sheet flow layer under non-repeatable wave conditions in prototype-scale laboratory or
field studies, an in-situ multi-point sensor is needed. A first prototype of a
conductivity concentration profiler (CCP) was developed at the University of
Delaware by Puleo et al. [2010] and used a two-electrode approach for conductivity
measurement. The conductivity probe of this sensor, however, was relatively large,
causing flow disturbance and unacceptable scour. Furthermore, the two-electrode
approach and the associated circuitry returned accurate results in fresh water but not in
salt water owing to higher conductivity. A second version of the CCP was developed
by Lanckriet et al. [2013] from the ground up based on the lessons learned from the
first prototype.
2.2
Sensor Design
The CCP consists of an internal electronic circuit contained within a PVC
housing with an outer diameter of 48.3 mm and a detachable external conductivity
probe (Figure 2.1). The total length of the sensor is 450 mm. The conductivity probe
consists of a 6-layer FR-4 (woven fiberglass with an epoxy binder) printed circuit
15
board with dimensions 292.2 mm (length), 5.6 mm (width) and 1.6 mm (thickness).
Two single arrays of 32 electrode plates spaced 1 mm in the vertical are located on
both sides of the probe stem near the tip with dimensions 0.5 mm height and 3.1 mm
width (Figure 2.1 inset). Electrode plates are coated with 76 nm of immersion gold for
chemical passivation and abrasion resistance. The electrode plates on both sides of the
probe at each elevation are internally connected so that each measurement is
performed on the two sides of the probe reducing potential bias from probe
deployments slightly oblique to the dominant flow direction. The PVC housing and
probe stem are buried in the sand bed during deployment such that only a small
section of the probe is proud of the sand-water interface. Video observations under
laboratory and natural conditions have shown that the small cross-section of the
exposed probe generates little flow disturbance for flows parallel to and slightly
oblique to the sensor orientation. The electrode plates where conductivity
measurements occur are located 131.7 mm above the PVC housing (Figure 2.1) so that
the housing itself does not cause interference.
Abrasion by sand particles under wave action can damage the surface of the
electrode plates, reducing the reliability of the signal. Therefore, the conductivity
probe was designed as easily replaceable and expendable. The connection of the
internal circuitry to the probe is implemented by inserting the bottom end of the
conductivity probe into a PCI Express x1 edge connector on the internal electronic
circuit board. Water tight conditions are maintained using a PVC collar around the
probe neck just above the connector that is secured in place with an IP68 cable gland.
During the BeST field study, the probes were replaced after being exposed to active
sediment transport conditions spanning one high tide cycle.
16
Communication cable to
shore
32 electrode plates
32 mm
IP68 cable
gland
168 mm
Disposable conductivity
probe
IP68 cable
gland
Internal circuitry
PVC housing
Total length 450 mm
Figure 2.1: The CCP sensor. Inset: the 32 gold-plated electrodes used for the
conductivity measurements.
The CCP electronic components are comprised of a microcontroller, serial data
transceivers, conductance measurement circuit, and electrode selection multiplexers
(Figure 2.2). Each conductivity measurement is performed using four neighboring
electrode plates on the probe, named Force+, Sense+, Sense- and Force- respectively.
The conductance-measurement process is initiated with the plate drive control circuit
producing a plate drive voltage suiting the conductivity of the medium that may range
from fresh water in laboratory settings to saline water in natural environments. The
analog multiplexers then connect the four electrode plates to the conductancemeasurement circuit and a measurement is taken using a 10 bit analog-to-digital (A/D)
converter on the microcontroller. After the conductance measurement is completed,
the multiplexers connect the next four electrodes to the measurement circuit and the
measurement cycle is repeated. A profile of 29 conductivity measurements is
generated by multiplexing through the array of 32 electrodes.
The conductance measurement circuit (Figure 2.3) is similar to Li and Meijer
[2005] and consists of a pair of op-amps (U10a, U10b) that drive the outer force
17
Figure 2.2: Schematic of the CCP measurement procedure.
electrodes (Force+, Force-) with sufficient voltage to cause the inner sense electrodes
(Sense+, Sense-) to experience a voltage equal to the drive voltage (Drive+, Drive-).
Electric current flowing as a result of the drive signal is detected as a voltage
difference across the resistors R25 and R35 by instrumentation amplifiers U8a and
U8b and summed by op-amp U11 to produce the detected conductance signal Vsense.
The polarity of current detection is opposite between the U8a and U8b channels so
spurious signals induced into both channels cancel when summed together to improve
noise immunity. Low-pass filtering in the form of R4 + C10 and R3 + C4 assists in
rejection of higher-frequency noise created by other instruments operating in the water
nearby. The output Vsense, the conductivity signal related to sediment concentration
(see section III.A), is connected to the A/D converter input on the microcontroller.
The actual measurement process is more complex than the preceding
description due to baseline shifts in the detected conductance resulting from the use of
probe board electrodes for both driving current and sensing voltage during the
18
multiplexing cycle. The forcing current through the electrodes builds up an
electrochemical potential that would corrupt subsequent measurements using those
same surfaces and must be neutralized by a reverse current flow. The microcontroller
samples the detected conductance at four different times during each measurement
cycle and calculates the baseline shift rate, removing it from the reported conductance
to resolve this problem.
The calculated error is then fed into a proportional-integral digital controller,
the output of which determines the polarity and duration of the compensatory reverse
current used to remove the electrochemical potential from the electrodes. The
duration of the measurement and potential-removal cycle is dependent on the error
U8a
–
Vref
+
Drive–
+
U10a
R25
Force–
–
R3
Vsense
U11 +
–
C4
R4
R28
Vref
Sense–
R5
R6
+
Drive+
U10b
R35
Force+
–
R38
C10
Sense+
U8b
Vref
+
–
Figure 2.3: Circuit diagram of the CCP analog conductance measurement unit.
19
compensation by the proportional-integral controller and limits the fastest sampling
rate of the current sensor configuration to 8 Hz.
Measured sensor output is transmitted in real time to a shore-based computer
through a pair of RS-422 / RS-485 transceivers connected to a waterproof direct-burial
Ethernet cable passing through the IP68 cable gland at the base of the housing (Figure
2.1). System operational parameters such as sampling rate, drive voltage, and digital
controller coefficients can be adjusted through the shore-based computer prior to
installation or on-the-fly during a deployment to ensure correct and stable operation.
2.3
Sensor Calibration and Measurement Characteristics
2.3.1
2.3.1.1
Calibration
Existing Relationships between Concentration and Conductivity
The rationale behind using conductivity as a proxy for sediment concentration
is that both fresh and saline water have a high electric conductivity, while noncohesive sediment grains are essentially non-conductive. The conductivity of any
sediment-water mixture is thus a function of the volume fractions of the two phases,
and it is expected that this effect is most significant when the sediment phase occupies
a significant fraction of the measurement volume. The CCP was calibrated by
measuring the conductivity of sediment-water mixtures with a known sediment
concentration. Generating mixtures with high sediment concentrations of up to the
packed bed limit, which typically have a solids fraction of 0.6-0.65, is difficult, except
for the limiting cases of no sediment (zero volume fraction) and the packed sand bed
itself. Previous efforts made measurements using a set of known sediment
concentrations, either by adding a known mass of sediment to a vessel filled with a
20
liquid or by taking suction samples at the same location as the conductivity sensor
[Horikawa et al., 1982; Bakker et al., 1988; Dick, 1989; Ribberink and Al-Salem,
1992; Puleo et al., 2010]. Sediment suspension consisted of volume fractions less
than 0.4 leaving a gap in the calibration curve for higher volume fractions. In
addition, ensuring the homogeneity of the sediment suspension at these high volume
fractions is difficult.
The previous studies, with the exception of Dick [1989], assumed a linear
relationship between sediment volume fraction and conductivity of the form:
c = k (1 − σm /σf )
(2.4)
where c is the volume fraction of sediment, 𝜎𝑚 is the conductivity of the mixture, 𝜎𝑓 is
the conductivity of the fluid and 𝑘 is a calibration constant. The formulation (2.4) can
be obtained from the theoretical development by Landauer [1952] for binary metallic
mixtures where there is no correlation between the positions of the two phases, in the
limiting case of zero conductivity for one of the two media
σm /σf = 1 − 3/2 c
(2.5)
by substituting the theoretical factor 2/3 by the calibration constant k [Dick, 1989].
Another commonly used formula for the conductivity of porous media is Archie’s law
[Archie, 1942]:
1/𝐹 = σm /σf = ϕm = (1 − c)m
(2.6)
where F is the formation factor, 𝜙 = (1 − 𝑐) is the porosity of the medium and m is a
calibration factor.
The theoretical Bruggeman equation [Bruggeman, 1935] for insulating spheres
of mixed sizes in a conductive medium is identical to Archie’s law with m = 1.5. De
21
La Rue and Tobias [1959] measured the conductivity for glass spheres and sand in a
zinc bromide suspension and found that Archie’s law gave a good fit with 1.43 ≤ 𝑚 ≤
1.58 (correlation coefficient squared, r2 > 0.998), while Archie [1942] found 𝑚 =1.3
for clean sands. Similar results were found in additional numerical and experimental
studies [Lemaitre et al., 1988; Küuntz et al., 2000]. Jackson et al. [1978] found that 𝑚
depends on the grain shape, with 𝑚 =1.2 for spheres, 𝑚 =1.85 for shell fragments and
𝑚 =1.39-1.58 for marine sands. In summary, past theoretical and experimental work
suggests 1.3 < m < 1.58 for sands.
2.3.1.2
Experiments for CCP Calibration
Experiments were carried out with two natural sand samples: a fine silica sand
(S1; d50 = 0.12 mm, d16 = 0.08 mm, d84 = 0.17 mm) and a coarse sand (S2; d50 = 0.44
mm, d16 = 0.25 mm, d84 = 0.72 mm), where dn is the grain size for which n % is finer.
The densities of the sand samples were 2.64 kg/m3 for S1 and 2.63 kg/m3 for S2. The
calibration experiment was carried out using an aqueous solution of Lithium
Metatungstate (LMT), a liquid with a nominal density of 2.95 kg/m3. LMT was
diluted using water to attain a density of 2.65 kg/m3, creating a liquid in which the
sand was approximately neutrally buoyant. The diluted LMT allowed sand-liquid
mixtures with a known, homogeneous sediment concentration from zero to
approximately 0.5 to be generated. For concentrations above 0.5 the mixture
resembled a loosely packed sand bed and became increasingly heterogeneous,
rendering measurements unreliable. The calibration experiments undertaken are
similar to those by De La Rue and Tobias [1959] who used zinc bromide as the
suspension liquid.
22
A set of 29 known masses of S1 sand (24 for S2) were individually added to
the liquid and manually stirred with a glass stirring rod creating nominal
concentrations ranging from 0 to 0.51 volume fraction (0 g/L to 1353 g/L) for S1, and
0 to 0.42 (0 g/L to 1161 g/L) for S2. Conductivity measurements using the CCP
sensor were recorded at 4 Hz for 25 s for each concentration, yielding 100 samples.
The mixture temperature was also recorded throughout the experiment using a
mercury thermometer and was 22 +/- 0.5 °C. Only the inner 19 locations in the
vertical concentration profile were retained to avoid edge effects from the probe being
too close to either the free surface or the bottom of the vessel, leaving a total of 1900
conductivity estimates for each concentration. The 1900 data points were averaged to
find a representative conductivity measurement for each sediment concentration and
the standard deviation of the 1900 samples was calculated to provide an estimate of
the temporal and spatial variability of the conductivity in the sample. For the fine
sand S1, the grains were neutrally buoyant and the mixture appeared visually to be
homogenous throughout the experiment. For the coarser sand S2, particles were
observed to be slightly positively buoyant as the liquid density could not be matched
exactly with the specific density of the sand. To alleviate this issue, only the first 5 s
of each conductivity measurement for S2 were used to obtain a conductivity value,
while still using the inner 19 measurement points in the vertical profile, leading to 380
data points used for each representative conductivity measurement as opposed to
1900. The buoyancy effect was most significant for small sediment concentrations, as
the vertical rising of the grains was slowed down significantly due to hindered rising
or increased viscosity of the slurry at higher sediment concentrations. No significant
vertical motion of the grains was observed for sediment concentrations higher than
23
1
Linear fit
0.9
Coarse sand S2
Linear fit
0.9
Power-law fit
0.8
0.8
0.7
0.7
σm /σf
σm /σf
1
Fine sand S1
0.6
0.6
0.5
0.5
0.4
Power-law fit
0.4
(a)
0.3
0
(b)
0.1
0.2
0.3
c (-)
0.4
0.3
0.5
0
0.1
0.2
0.3
c (-)
0.4
0.5
Figure 2.4: Conductivity calibration for fine sand S1 (a) and coarse sand S2 (b).
Vertical bars represent 1 standard deviation on either side of the mean of
the 1900 (a) or 380 (b) samples used for each of the representative
conductivity measurements. The gray and black dashed lines are least
squares fits using the linear law (2.4) and power law (2.6) respectively.
0.27. The CCP measurements indicate an obvious decreasing trend in conductivity as
a function of sediment concentration for both the fine sand (Figure 2.4a) and coarse
sand (Figure 2.4b). The small variability, indicated by the vertical bars at each
concentration in Figure 2.4, signifies the homogeneity of the mixtures. No obvious
trend in temporal variability was found in the time series of individual sensor
Table 2.1:
Results of calibration experiments.
Fine sand
(S1)
Coarse sand
(S2)
d50 (mm)
0.12
0.44
Linear law
(2.4)
k
r2
0.81
0.997
0.86
0.987
Power law
(2.6)
m
r2
1.36
0.998
1.19
0.985
24
channels. Measurements show excellent correlation to both a linear relationship
[equation (2.4)] and a power-law relationship [equation (2.6)], determined using a
least-squares fit. The power law calibration constants m are 1.36 for S1 and 1.19 for
S2, near the range of values reported in the literature (Table 2.1)
2.3.2
2.3.2.1
Measurement Volume Quantification
Theory
CCP measurements are influenced by the size of the measurement volume,
which is mostly determined by the electric field emitted by the electrodes. Four
electrodes are used for each conductivity point measurement. Due to the finite size of
the rectangular electrode plates (Figure 2.1 inset), the voltage field around the probe is
three dimensional and cannot be approximated by an analytic expression based on
point electrodes. Instead, a finite differences model was used to study the voltage
potential and current fields around the CCP probe, in an effort to estimate the
horizontal extent of the measurement volume and the amount of smoothing that occurs
when measuring a varying vertical conductivity profile.
During the design of the CCP sensor, oscilloscope measurements verified that
the duration of each electric pulse emitted by the force electrodes is longer than the
relaxation time of the surrounding environment to equilibrate to the forced voltages.
Therefore, the electric field around the sensor can be calculated using the laws of
stationary electromagnetism. The continuity equation for a stationary electric current
is
∇ ⋅ 𝐽⃗ = 0
25
(2.7)
where 𝐽⃗ is the current density field. In a non-homogeneous medium 𝐽⃗ can be
calculated as
𝐽⃗ = 𝜎𝐸�⃗ = −𝜎∇𝑉
(2.8)
where 𝐸�⃗ is the electric field, 𝜎 is the electric conductivity and 𝑉 is the voltage
potential. Combining (2.7) and (2.8) yields
∇ ⋅ (𝜎∇𝑉) = 0
(2.9)
Equation (2.9) reduces to the well-known Laplace equation for a homogeneous
conductivity field. The following assumptions were made to model the current field:
1.
The thickness of the CCP probe is negligible. As the probe has
electrode plates on both sides that are internally connected, the entire
geometry is symmetrical around the plane of the electrode plates.
2.
To reduce computation time, individual sand grains were not resolved.
Instead, the sand-water mixture was modeled as a single medium with a
conductivity field that varies only on the scale of the sheet thickness as
a function of the sediment concentration. Furthermore, the
conductivity profile was assumed to vary only in the vertical direction:
𝜎 = 𝜎(𝑧).
The axis orientation for the model is displayed in Figure 2.5a. Since the
geometry is symmetrical around the x-z plane and the y-z plane, only 2 octants of the
space around the probe were modeled and (2.9) was solved on a domain with 0 mm ≤
x ≤ 12.7 mm, 0 mm ≤ y ≤ 12.7 mm and -8.13 mm ≤ z ≤ 8.13 mm. Equations (2.4-6)
are linear and only relative voltages are important. The Force+ and Force- electrodes
were modeled as boundary conditions with constant voltages of +1 V and -1 V
respectively. Floating electrode plates such as the Sense+ and Sense- electrodes and
the nearby unused electrodes were additional model boundary conditions where the
voltage on each floating electrode is constant since the electrode plates are far more
conductive than the surrounding medium and there is no net current flux into the
electrodes. The epoxy face of the probe was modeled as a no-flux boundary
26
condition. The outer edges of the domain were also modeled as no-flux boundary
conditions and were far enough away from the active electrodes as to not influence the
current field.
The voltage field V(x,y,z) was discretized on a grid with a constant spacing
Δ𝑥, Δ𝑦, Δ𝑧 and the conductivity field 𝜎(𝑧) on a grid that is staggered by one half of the
grid spacing in the z-direction, but collocated with the V grid in the x and y direction.
Using Gauss-Seidel iteration with a stability criterion analogous to the classical
Laplace equation, the solution was iterated as:
𝜔 �
𝜎𝑐,𝑗
𝜎𝑚𝑎𝑥
𝑛𝑒𝑤
𝑉𝑖,𝑗,𝑘
= 𝑉𝑖,𝑗,𝑘 +
Δ𝑦 2 Δ𝑧 2
�2(Δ𝑥 2 Δ𝑦 2+Δ𝑥 2 Δ𝑧 2 +Δ𝑦 2 Δ𝑧 2 ) �𝑉𝑖−1,𝑗,𝑘 − 2𝑉𝑖,𝑗,𝑘 + 𝑉𝑖+1,𝑗,𝑘 � +
Δ𝑥 2 Δ𝑧 2
2(Δ𝑥 2 Δ𝑦 2 +Δ𝑥 2 Δ𝑧 2 +Δ𝑦2 Δ𝑧 2 )
Δ𝑥 2 Δ𝑦2
2(Δ𝑥 2 Δ𝑦2 +Δ𝑥 2 Δ𝑧 2 +Δ𝑦 2 Δ𝑧 2 )
�𝑉𝑖,𝑗−1,𝑘 − 2𝑉𝑖,𝑗,𝑘 + 𝑉𝑖,𝑗+1,𝑘 �� +
𝜎𝑠,𝑗−1 𝑉𝑖,𝑗,𝑘−1 −�𝜎𝑠,𝑗−1 +𝜎𝑠,𝑗 �𝑉𝑖,𝑗,𝑘 +𝜎𝑠,𝑗 𝑉𝑖,𝑗,𝑘+1
𝜎𝑚𝑎𝑥
� (2.10)
where the index 𝑐 stands for the collocated 𝜎 grid and 𝑠 stands for staggered grid and
𝜔 = 1.3 is an over-relaxation factor that accelerates convergence. The system was
considered to have converged when
1
and
𝜖 = max��𝑉
𝑖,𝑗,𝑘 ��
2
𝑛𝑒𝑤
∑𝑖,𝑗,𝑘�𝑉𝑖,𝑗,𝑘
− 𝑉𝑖,𝑗,𝑘 � < 10−4
𝐼 + −𝐼 −
��
𝐼−
� − 1� < 0.05
(2.11)
(2.12)
where 𝜖 is the relative error and 𝐼 + and 𝐼 − are the magnitudes of the current through
the Force+ and Force- electrodes respectively and | | indicates magnitude. To further
accelerate convergence, a multigrid approach was adopted where the model was
solved subsequently on 6 increasingly finer grids (Table 2.2). The voltage field of the
27
Table 2.2:
Grid spacings in the multigrid approach
Grid
Grid 0
Grid 1
Grid 2
Grid 3
Grid 4
Grid 5
Grid spacing
Δ𝑥 = Δy =1.046 mm; Δ𝑧 = 0.508 mm
Δ𝑥 = Δ𝑦 = Δ𝑧 = 0.508 mm
Δ𝑥 = Δ𝑦 = Δ𝑧 = 0.254 mm
Δ𝑥 = Δ𝑦 = Δ𝑧 = 0.169 mm
Δ𝑥 = Δ𝑦 = Δ𝑧 = 0.127 mm
Δ𝑥 = Δ𝑦 = Δ𝑧 = 0.102 mm
Nodes
6 468
22 308
171 666
560 272
1 326 130
2 556 036
next finer grid was initialized using a linear interpolation of the previous coarser grid.
Grids 0 to 4 were only used to speed up the computation. Results from the finest grid,
with spacing 0.102 mm in all directions, were used for subsequent analysis.
2.3.2.2
Lateral Extent of Current Field
When the CCP probe is deployed upright through the sand bottom, the lateral
extent of the CCP measurement volume indicates how far away from the sensor the
measurement is influenced in the plane normal to the stem of the probe. This is
important for calculating the quantity of sand grains in the measurement volume. If
the measurement volume only contained a small number of sand grains, the CCP
would likely output a “binary signal”: high conductivity when there is no sand present
in the measurement volume, and low conductivity when a small number of sand grains
are present. Additionally, the lateral extent of the measurement volume indicates how
far a CCP sensor must be placed from any obstacles such as the side wall of a flume or
other sensors, to avoid disturbing the conductivity measurement.
The electric field was calculated for a constant conductivity, 𝜎, in the entire
domain to determine the lateral extent of the measurement volume. The conductivity
of the medium was set to 1 since the model equations (2.4-6) are linear and only
relative currents were considered. Then the current density 𝐽𝑧 oriented normal through
the x-y plane was computed, half way between the Force+ and Force- electrodes where
28
the lateral extent of the current field was the largest (Figure 2.5a). Figure 2.5b shows
the 𝐽𝑧 field normalized by the maximal current density in the plane. The current
density decays away from the sensor with the bulk of the current occurring near the
electrodes. The black curve in Figure 2.5 indicates where the current density has
decayed to 1 % of its maximum value. By integrating the current density through the
Figure 2.5: (a) Location of the x-y plane where the normalized current field is
computed. (b) Normalized current field through x-y plane indicating
lateral extent of the measurement volume. The black curve in (b)
indicates where the current field is 1 % of its maximum value.
29
surface delineated by this 1 %-curve, 88 % of the total current passes through this
surface. Therefore, the 1 %-curve is defined as the lateral extent of the measurement
volume. The measurement volume extends from -8.7 mm ≤ x ≤ 8.7 mm and -8.4 mm
≤ y ≤ 8.4 mm. As the probe is 5.6 mm wide, the measurement volume extends
approximately 1.5 probe widths away from the center line of the probe. However, this
lateral extent of measurement volume estimate is conservative since most of the
current flows much closer to the electrodes than the 1 % curve (Figure 2.5). The
linearity of the model equations also means that the relative current density field, and
thus the measurement volume, is independent of the absolute conductivity value. As a
result, the measurement volume is identical for measurements in both high and low
conductivity environments such as clear water and a packed sand bed.
2.3.2.3
Vertical Extent – Profile Smoothing
It is expected that measured sheet flow layer thicknesses will appear larger
than the real sheet flow layer thickness through smoothing of the vertical
concentration profile as a result of the finite extent of the measurement volume. The
electric field was again simulated with the finite difference model, this time for a
piecewise linear transition in concentration between a simulated “compacted sand”
with a volumetric sediment concentration of 0.644 [Bagnold, 1966b], and “clear
water” with a volumetric sediment concentration of 0. The piecewise linear
concentration profile was chosen as a simplified, representative shape of the
conductivity profile in the sheet layer, which may have an arbitrary shape. The
corresponding conductivity field was computed using Archie’s law (2.6) with m =
1.5. Since Archie’s law yields an approximately linear relation between concentration
30
and conductivity, this corresponds to an approximately piecewise linear transition in
the conductivity profile.
The conductivity profile was shifted vertically by increments of 0.5 mm
through the computational domain and the virtual sensor output was calculated for
each shift. Shifting the conductivity profile in the domain is equivalent to measuring a
fixed conductivity profile with electrodes at different elevations, as would occur for
actual CCP sensor measurements. Figure 2.6a shows an example conductivity profile
with the thickness of the prescribed conductivity transition of 4.0 mm (dashed line).
10
8
6
6
4
4
2
2
0
-2
0
-2
-4
-4
-6
-6
-8
-8
-10
0
(b)
8
z (mm)
z (mm)
10
(a)
0.2
0.4
0.6
σ / σf (-)
0.8
-10
1
0
0.2
0.4
0.6
σ / σf (-)
0.8
1
Figure 2.6: (a) Conductivity profile for a 4.0 mm prescribed conductivity transition.
Dashed line is the prescribed conductivity profile of the medium;
triangles indicate the 5 % and 95 % cutoff for the prescribed profile. The
solid line is the simulated conductivity profile for the CCP; squares
indicate the 5 % and 95 % cutoff for the simulated profile. (b)
Comparison of simulated CCP results (solid black line) with
experimental CCP measurements (gray dots) across a sand-water
interface.
31
The profile as it would be measured by a simulated CCP sensor is smoothed (solid
line) and the thickness of the transition layer is larger than the prescribed thickness.
Numerical simulations were validated using measurements across a sand-water
interface. A CCP was placed in a vessel containing water. Coarse sand (S2) was
added and allowed to settle under gravity until the sand-water interface was located at
roughly the middle of the profiling region of the probe. Fifteen measurements were
taken, each time raising the sensor by 0.2 mm between measurements using a
manually-controlled stepper to analyze sensor smoothing at sub-millimeter resolution.
Model simulations were performed with the conductivity in the fluid and in the sand
bed equal to the measured values (Figure 2.6b). As it can be expected that the grain
packing density near the top of the sand bed will be less than the packing deeper in the
sand bed, the sand-water interface was modeled as a piecewise linear concentration
profile in which the transition thickness from a fully packed bed to clear water was set
to 2d50 = 0.88 mm. The simulated CCP output was found to be relatively insensitive
to this value due to its vertical smoothing. The shape of the simulated CCP
measurements agrees well with the real measurements (r2 = 0.985), illustrating the
skill of the developed model (Figure 2.6b).
Similar simulations as in Figure 2.6a were conducted for conductivity
transitions with varying prescribed thicknesses ranging from 0 mm to 14 mm. The top
and bottom of the sheet flow layer were defined using a curve-fitting technique to find
a ‘shoulder’ point in the concentration profile (indicative of the bottom of the sheet
flow layer) that provided the best results for defining sheet flow layer thickness under
field conditions (described in Section 4.3.1). For each simulated conductivity
transition, the sheet flow layer thickness was determined using the curve-fitting
32
method for both the prescribed conductivity/concentration profile (before smoothing)
and the conductivity/concentration profile as measured by the simulated CCP
(including sensor smoothing). Sheet thicknesses determined after smoothing by a
simulated CCP sensor (Figure 2.7, black line with triangles) and before smoothing
(solid black line) co-vary but the profile smoothing causes an overprediction of
simulated sheet thicknesses. For sheet thicknesses (not taking smoothing into
account) of less than 5 mm (corresponding to an estimated sheet thickness of 5.5 mm
including smoothing), the simulated sheet thickness is dominated by the smoothing
effect and is insensitive to the prescribed sheet thickness. For sheet thicknesses
exceeding 5 mm, the simulated CCP signal resembles the prescribed sheet thickness.
A correction formula was developed that converts sheet thicknesses estimated from
CCP measurements (including smoothing) to the correct sheet thicknesses:
𝛿𝑠𝑒𝑛𝑠
𝛿𝑟𝑒𝑎𝑙
=
1
2
713⋅103 𝛿𝑠𝑒𝑛𝑠
−6024𝛿𝑠𝑒𝑛𝑠 +21.9
+ 1,
(2.13)
with 𝛿𝑟𝑒𝑎𝑙 and 𝛿𝑠𝑒𝑛𝑠 the real thickness and measured thickness by a CCP sensor,
respectively, expressed in meters.
A conservative manner of interpreting sheet flow thicknesses from CCP
laboratory or field experiments is to discard measurements with an uncorrected
measured sheet thickness of less than 5.5 mm, which corresponds to a real sheet
thickness of 5 mm according to the correction formula (2.13). Measurements with real
sheet thicknesses of 5.0 mm or more (corresponding to uncorrected thicknesses of 5.5
mm or more) are reliable (see Discussion), and their actual thicknesses are corrected
using (2.13). It is noted that the simulation displayed in Figure 2.6a is near the limit of
reliable sheet thickness. This simulation is therefore near the worst case for the error
in sheet thickness due to CCP smoothing.
33
Initial analysis of the sensor smoothing effect [Lanckriet et al., 2013] defined
the bottom of the sheet layer as the elevation where the conductivity exceeds the
minimum conductivity plus 5 % of the difference in conductivity between the
maximum and minimum conductivity, and the top of the smoothed transition layer as
the elevation where the conductivity exceeds the minimum conductivity plus 95 % of
this difference (Figure 2.6). Subtracting the 5 % level from the 95 % level then yields
Thickness based on simulated CCP sensor (incl. smoothing) (mm)
an alternate estimate of the measured sheet layer thickness (Figure 2.7, blue line). A
14
12
10
8
6
4
2
0
0
2
4
6
8
10
12
14
Thickness based on perfect non-smoothing sensor (mm)
Figure 2.7: Sheet thickness as measured by a simulated CCP sensor compared to
sheet flow layer thickness determined without sensor smoothing. Black
line with triangles: sheet thickness determined using curve-fitting
technique (Section 4.3.1). Red line with circles: sheet thickness
determined via 5%-95% cutoff of conductivity profile. Blue line with
squares: sheet thickness determined via 5%-95% cutoff of conductivity
profile for miniaturized CCP sensor (Section 2.4.1). Black solid line is
line of perfect agreement.
34
correction factor was also developed for this case and is mentioned here only for
completeness:
𝛿𝑠𝑒𝑛𝑠
𝛿𝑟𝑒𝑎𝑙
=
1
2
127⋅103 𝛿𝑠𝑒𝑛𝑠
−94.1𝛿𝑠𝑒𝑛𝑠 −2.07
+1
(2.14)
Additionally, the sediment volume mobilized within the sheet flow layer, or
volumetric sheet load, can be determined as
𝑧
Φ = ∫𝑧 𝑡 𝑐 𝑑𝑧
(2.15)
𝑒
where 𝑧𝑒 and 𝑧𝑡 are the elevation of the bottom and top of the sheet flow layer,
respectively. The sediment mass mobilized in the sheet layer, or sheet load, is then
defined as
(2.16)
C = 𝜌𝑠 Φ,
where 𝜌𝑠 = 2650 𝑘𝑔/𝑚3 is the sediment mass density. A correction formula was
also developed to correct the volumetric sheet load for the smoothing effect:
Φsens
Φreal
=
1
713⋅103 Φ2𝑠𝑒𝑛𝑠 −6024Φ𝑠𝑒𝑛𝑠 +21.9
+1
(2.17)
where Φ𝑟𝑒𝑎𝑙 and Φ𝑠𝑒𝑛𝑠 are the real volumetric sheet load and measured volumetric
sheet load (including smoothing), respectively.
2.4
2.4.1
Discussion
Sensor Design
Several factors affected the design of the in situ probe, including the
minimization of flow disturbance and scour around the probe, mechanical strength for
field deployment, and an optimal size of the measurement volume. With cross section
dimensions of only 1.6 mm x 5.6 mm, flow disturbance and bed scour was found to be
minimal during initial testing in a wave flume and during deployments on a natural
35
beach for flow parallel to the sensor orientation. For flows oblique to the sensor, eddy
shedding will occur causing flow disturbance behind the sensor meaning sensor
orientation during deployment is critical. The mechanical strength of the probe, made
of woven fiberglass with an epoxy binder, is considerable. Additionally the probe is
supported through its burial in the sand bed with only the top 10 - 40 mm exposed. As
a result, no probes have failed mechanically under wave action during the lab and field
trials.
The optimal size of the measurement volume should be large enough to
contain a sufficient number of sand grains in order to make a representative
measurement of the sediment concentration, and small enough to minimize smoothing
of the vertical concentration profile. The measurement volume can be approximated
by an ellipsoid with semi-axes lengths x: 8.7 mm; y: 8.4 mm (as determined by the
lateral extent of the measurement volume; section III.B.2); z: 3.5 mm (the smallest
sheet thickness that is resolved by the sensor based on a 5% and 95% cutoff of the
conductivity profile), giving a volume of 1071 mm3. For volumetric concentrations
between 0.08 to 0.51 and sand grains approximated as spheres with diameter of 0.33
mm (d50 of sand during the Beach Sand Transport experiment, Chapter 3), the
measurement volume would contain 4.6⋅103 to 2.9⋅104 sand grains.
This large
number of sand grains is sufficient to provide a representative measurement of the
sediment concentration. For comparison, an optical backscatter sensor (OBS) for
suspended sediment concentration has a measurement volume of 1-20⋅103 mm3
[Downing, 2006]. For volumetric sediment concentrations of 0.03 (100 g/L) in
energetic surf zones, an order of magnitude smaller than the typical range of
concentrations measured by the CCP, the OBS measurement volume contains 2⋅103 to
36
4⋅104 sand grains, of the same order as the CCP measurement volume. In contrast, a
fiber optic backscatter sensor (FOBS) has a measurement volume of only 25 mm3,
corresponding to 50 grains at a volumetric sediment concentration of 0.03. Thus,
longer averaging times are needed to avoid noise at low concentrations when using a
FOBS [Downing, 2006].
The vertical smoothing of the CCP measurements means the sensor cannot
detect sheet flows less than 5 mm thick. A correction factor must be applied to
analyze larger sheet thicknesses. Following the criterion of Nielsen [1992] for the
onset of sheet flow (𝜃 > 1.0) and the estimate of sheet thickness under stationary flow
[equation (1.3)], sand grains with d50 = 0.35 mm will form a sheet layer with thickness
3.5 mm at the onset of true sheet flow (absence of bed forms). Therefore, in flows
where the hydrodynamic forcing exceeds the threshold for true sheet flow, the theory
predicts a sheet flow layer with a thickness that can almost be resolved by the CCP.
Near-bed sediment transport can also be dominated by granular interaction occurring
at lower Shields numbers [Horikawa et al., 1982]. Sheet layer thicknesses in these
flows are not resolved by the CCP. Sheet thicknesses recorded during field
observations (Section 4.2) routinely exceeded 10 mm during both uprush and
backwash. At these thicknesses, the smoothing effect is small (Figure 2.7). In
summary, the measurement volume of the CCP is large enough to contain a sufficient
number of grains for a representative concentration measurement, and small enough to
resolve sheet flow in most practical applications. Moreover, the detailed study of the
smoothing effect using numerical simulations allows the user to address vertical
smoothing using a correction factor, rather than assuming the inherent smoothing does
not affect sheet thickness estimates.
37
Further miniaturizing the CCP probe could reduce the smoothing effect but this
has to be weighed against increased manufacturing costs. As the sediment
concentration in the sheet layer has a sharp gradient in the vertical but is
approximately uniform in the horizontal, positioning the four electrodes needed for a
conductivity measurement in a horizontal row would reduce the vertical extent of the
measurement volume and thus the smoothing effect. On the other hand, each point
measurement in the concentration profile would then require a separate row of four
electrodes, increasing the number of electrodes and electrical routes in the probe by a
factor of four. Measurement volume simulations were conducted for a probe believed
to be near the limit of current manufacturing capabilities to estimate the maximum
possible reduction of the measurement smoothing. The electrode plates on this
idealized probe were modeled as squares with dimensions 0.5 mm (the width of the
rectangular electrode plates on the current probe), placed in horizontal rows of four
electrodes with a spacing of 1 mm. The vertical profile smoothing by this probe based
on the 5% and 95% conductivity cutoff (blue line, Figure 2.7) would be smaller than
the current probe and sheet thicknesses down to 1.5 mm could be resolved. An
additional benefit of the horizontal lay-out is that electrodes would maintain their
function as either Force or Sense electrodes during the multiplexing process using this
lay-out, negating the build-up of electrochemical potential. As a result, the CCP could
sample at higher frequencies without the need for an electrochemical potential
removal step as described in section 2.2.
2.4.2
Sensor Calibration
A range of sediment volume fractions from 0 to 0.51 (0 to 0.44) was tested for
fine sand S1 (coarse sand S2) during the calibration experiments. Over this range,
38
conductivity was reduced from the clear-water conductivity by a factor of 2.6 (2.2).
Extrapolating this result using the power law [equation (2.6)], conductivity would be
reduced by a factor 4.0 (3.1) if a measurement was made in a fully compacted bed
with a volume fraction 0.644. This large reduction in conductivity and the correlation
with calibration laws justify the use of conductivity as a measurement method for
sediment concentration.
Many prior studies of sheet flow in coastal environments used a linear
relationship between conductivity and sediment concentration [e.g., Horikawa et al.,
1982; Bakker et al., 1988; Ribberink and Al-Salem, 1992; McLean et al., 2001;
Dohmen-Janssen and Hanes, 2002, 2005; O’Donoghue and Wright, 2004a]. Testing
both the linear and power-law relationships in this study revealed that both are equally
adept at describing the experimental data as determined by the square of the
correlation coefficient. The assumptions underlying the theoretical Bruggeman
[1935] equation, equivalent to the power law, are more suited to the case of sediment
grains suspended in water than the assumptions underlying the derivation by Landauer
[1952], equivalent to the linear law. In addition, there is more numerical and
empirical evidence in the literature pointing toward a power-law relationship for the
conductivity of sand-water mixtures [Archie, 1942; De La Rue and Tobias, 1959;
Jackson et al., 1978; Lemaitre et al., 1988; Küuntz et al., 2000]. Hence, a power-law
relationship [equation (2.9)] is adopted as the method to calibrate the CCP sensor.
The calibration factor, m, for a power-law relationship between electrical
conductivity and sediment concentration was found to be 1.36 and 1.19 for fine (S1)
and coarse (S2) sediments respectively. The value for S1 falls within the range of
those reported in previous studies (m = 1.3 - 1.6) while the value for S2 falls slightly
39
outside this range. A possible cause for the smaller m value for coarse sand is that the
sand-liquid mixture may not have been perfectly homogenous. Additionally, fines
were removed from the S2 sand sample using a woven polyester mesh with 0.025 mm
opening size during a previous study. The lack of fines in the sand-liquid mixture may
have altered the pore structure and the conductivity response. Nevertheless, CCP
sensor data show a strong response to the sediment concentration for both samples and
the calibration laws describe measurements accurately (r2 > 0.98). Less guidance
exists in the literature on the value of the calibration coefficient k in a linear fit model.
Ribberink and Al-Salem [1992] obtained a value k = 1.16 for sand with d50 = 0.21 mm
(median grain diameter between S1 and S2 in the present study), larger than the values
found here (k = 0.81 and 0.86 for S1 and S2 respectively). Fitting a power law
through the linear relationship used by Ribberink and Al-Salem [1992] yielded m =
0.83, in contrast with existing literature for the power-law relationship. A possible
explanation is that the calibration by Ribberink and Al-Salem [1992] may have been
influenced by their conductivity measurements in a non-moving sand bed, where a
small error in the measured porosity values may have led to large differences in the
calibration coefficient. Dick [1989] obtained k = 1.0 for acrylic particles with d50 =
0.5 mm and 𝜌 = 1.141 kg/m3, in between the value obtained by Ribberink and Al-
Salem [1992] and the values obtained in the present study. Their calibration value
may have been affected by the shape of the acrylic particles which influences the
calibration coefficient [Jackson et al., 1978].
Laboratory calibration measurements were continued until approximately the
loosely packed bed limit with a volume fraction of roughly 0.5. For higher sediment
concentrations, the sediment-water mixture could no longer be kept homogeneous. In
40
practical situations, however, the CCP is used to make concentration measurements up
to the packed bed limit with a volume fraction of 0.644. Since Archie’s law has also
been validated for consolidated sandstone materials with solids volume fractions of
0.7 – 0.9 [Archie, 1942; Adler et al., 1992], it may be assumed that the calibration
laws can be extrapolated to the bed packed limit.
To perform a sensor calibration in a practical case such as a field study, two
coefficients in equation (2.6) need to be determined: the calibration factor m, which
describes the reduction in conductivity by the presence of sediment, and the fluid
conductivity 𝜎𝑓 . The factor m is solely determined by sediment characteristics such as
shape and size of the sediment and is therefore usually constant during a field
campaign. The fluid conductivity may vary throughout a field campaign due to
variations in water temperature and salinity. Since m is independent of the fluid
conductivity, calibration measurements using fresh water, saline water or LMT
solution must yield the same m for the same sediment source.
Performing a full sensor calibration using a heavy liquid for every field site is
difficult because of the cost of the heavy liquid. Instead, a reasonable field calibration
scheme is as follows: the fluid conductivity 𝜎𝑓 is determined by taking measurements
high in the water column, where no or negligible sediment is present, and must be
updated frequently to account for variations in fluid conductivity. The calibration
factor m is then determined by measuring the reduction in conductivity in a packed
sediment bed with a known sediment volume fraction. Values for m obtained from
measurements throughout the field campaign can then be averaged to yield a single,
more robust value. This approach was used for the field results presented in Chapter
41
4, yielding a calibration factor m of 1.41, within the range of values presented in this
study and in the literature.
2.5
Conclusions
A novel sensor was developed to measure the vertical profile of sediment
concentration under sheet flow conditions. The sensor consists of a detachable probe
with 32 plate electrodes spaced 1 mm in the vertical and an internal electronic circuit.
The conductivity measurement is performed using a four-electrode approach, and a
29-point profile with 1 mm resolution is obtained by multiplexing through the
electrode array. The conductivity response to sediment concentration was validated
by measurements with known sediment masses in a Lithium Metatungstate (LMT)
solution and agrees well with existing power-law (Archie’s law) and linear
relationships between electrical conductivity and sediment concentration. Since a
power-law relationship is more supported by existing literature, this relationship was
adopted as the calibration curve for the sensor and calibration coefficients m = 1.36
and 1.19 were found using a fine and coarse sand respectively.
The measurement volume of the sensor was studied using numerical
simulations of the electric field around the conductivity probe. The horizontal extent
of the measurement volume is estimated as extending 8.7 mm and 8.4 mm (1.5 times
the sensor width) from the center of the sensor along the two principal horizontal axes.
The vertical extent of the measurement volume causes a smoothing of the vertical
concentration profile. Sediment concentration profiles with a sheet thickness of less
than 5 mm are not resolved by the sensor. A correction factor was developed to
account for the smoothing effect at larger sheet thicknesses.
42
Chapter 3
THE BEACH SAND TRANSPORT FIELD STUDY:
EXPERIMENT CONDITIONS
Data presented in this work were collected during the Beach Sand Transport
(BeST) field experiment, conducted in Perranporth, UK in October 2011. This study
was conceived in response to the First International Workshop on Swash-Zone
Processes, which identified comprehensive field measurements as a key need for
future swash zone research [Puleo and Butt, 2006]. The experiment was conducted
on October 9–15, 2011 as a collaboration between Plymouth University (UK), the
University of New South Wales (Australia) and the University of Delaware (USA)
with the aim of providing a comprehensive dataset of hydrodynamics, sediment
transport and morphodynamics in the swash zone of a natural beach. A detailed
overview of the study site, wave conditions and deployed instruments was also given
by Puleo et al. [2014b].
3.1
Field Site
The field experiment was conducted on Perranporth beach, Cornwall, United
Kingdom (Figure 3.1). Perranporth beach is a macrotidal, dissipative beach facing
west-northwest and is enclosed between two headlands (Droskyn and Ligger Points)
that are separated by approximately 3.5 km.
43
Figure 3.1: Location of the BeST field site in Perranporth.
Measurements were taken for approximately 3 hours around high tide for 10
consecutive tidal cycles around spring tide. Figure 3.2a and Table 3.1 show the
predicted tidal level during this period. These particular tidal sequences, with a mean
tide range of 5.43 m, were chosen because the high tides were all centered around 3 m
Ordnance Datum Newlyn (ODN). The significant wave height (Figure 3.2b) and
spectral peak period (Figure 3.2c) were measured in approximately 10 m water depth
Table 3.1:
Offshore forcing conditions during the BeST field study.
Tide number
1
Date (day in October 2011) 10
Hs (m)
2.46
Tp (s)
9.9
Max. tide level (m Ordnance 3.01
Datum Newlyn)
2
11
2.28
11.7
2.93
3
11
2.03
11.0
3.11
44
4
12
1.47
10.2
3.03
5
12
1.52
15.2
3.15
6
13
1.49
13.0
3.08
7
13
1.27
11.0
3.12
8
14
0.96
10.5
3.06
9
14
0.56
9.4
3.03
10
15
0.83
10.6
2.98
02
Elev. ODN (m)
01
3
03
04
05
06
08
07
10
09
a)
2.5
2
s
H (m)
3
1
b)
15
p
T (s)
20
c)
10
Direction (°)
5
300
250
d)
10
11
13
12
Day in October 2011
14
15
16
Figure 3.2: Offshore wave conditions for the BeST study. a) tide level. b)
significant wave height. c) spectral peak period. d) wave direction.
Horizontal dashed line is shore normal incidence. Gray shading in all
plots indicates the sampling period.
by a Datawell Directional Waverider buoy (50°21′11.34″N, 5°10′30.11″W; Channel
Coastal Observatory; www.channelcoast.org). Significant wave height at the
beginning of the study exceeded 2.5 m and decreased over the remaining tidal cycles
to around 0.5 m during the last high tide observed. The spectral peak period was
approximately 10 s for the first four tidal cycles before increasing to 15 s before the
fifth tidal cycle. The spectral peak period then decreased over the remainder of the
experiment until it was again nearly 10 s during the last tidal cycle. The offshore peak
45
wave direction varied by up to 30° relative to shore normal (the dashed horizontal line
in Figure 3.2d). Wave direction showed variability throughout the tidal cycle,
implying that tidal effects at the location of the buoy played an important role in the
depth influence on wave refraction. However, the wave obliquity in the surf and
swash zones did not seem to exhibit this much spread about shore normal.
Figure 3.3: a) Beach profiles measured prior to high Tide 7 on October 13, 2011,
along the central, north, and south transects: the horizontal dotted line
identifies the mean high water level, the solid gray horizontal line
indicates the cross-shore extent of the scaffold rig, and the black square
the cross-shore location of the main instrument bar. b) median grain size
within the extent of the scaffold rig as a function of cross-shore distance.
46
A 45-m-long scaffold frame was installed near the high tide line for mounting
sensors (Section 3.2). A local right-handed coordinate system was established with
cross-shore distance, x, increasing onshore and alongshore distance, y, increasing to
the north. The beach profile down the center of the scaffold frame, and 25 m on either
side of the frame, was surveyed using an electronic total station before and after each
tide. Beach profiles measured prior to high tide 7 on October 13, 2011, are shown in
Figure 3.3a. Beach profiles showed little alongshore variation and the slope near the
mean high water line was roughly 1:45 for all three profiles. Surface sediment
samples were collected along the central profile from x = −75.7 to −29.0 m at roughly
1 meter intervals. The median grain size, 𝑑50 , was determined using a settling tube.
𝑑50 near the main instrument bar was 0.33 mm, but showed a slight coarsening trend
with increasing onshore distance (Figure 3.3b).
Repeated surveying of the beach profile showed a consistent steepening of the
landward portion of the beach face throughout the experiment (Figure 3.4a). The
cumulative elevation change indicated erosion of the order of 0.1 m seaward of the
main instrument bar (the white dashed line in Figure 3.4a). Corresponding accretion
of approximately 0.1 m was observed near the landward end of the scaffold frame
where the landward and seaward ends of the scaffold frame are identified by the
horizontal dashed black lines. Cross-sectional area changes based on pre- and posttide
centerline cross-shore profile surveys were integrated across the profile and used to
infer the net cumulative sediment flux per beach width required to cause the observed
profile change. Cumulative sediment flux estimates varied from about 0.5 to 1.2 m2
(Figure 3.4b). There was a flux increase in association with the increase in wave
period around
47
Figure 3.4: Morphological variability during the BEST field study. a) Cumulative
elevation change for each tide: horizontal black lines denote seaward and
landward edges of the scaffold frame and horizontal white line indicates
the location of the main instrument bar. b) Cumulative volumetric flux
required to cause measured elevation changes determined from pre- and
post-tide beach profile surveys.
high tides 6 and 7 even though the significant wave height was smaller than earlier in
the study period.
3.2
Instrumentation
Sensors were mounted on a 45-m-long scaffold frame installed near the high
tide line (Figure 3.5a). Forty-five ultrasonic distance meters (UDMs; Massa M300/95)
were installed at an elevation approximately 1 m above the bed at 1 m intervals
48
spanning the length of the frame. Each UDM measured the free surface when the area
below it was immersed or the sand elevation when the area below it was exposed.
Thus, the sensors can be used to obtain cross-shore time series of water depth and
morphological change in the swash zone [Turner et al., 2008]. UDMs were sampled
at 4 Hz. A SICK lidar line scanner (SICK LMS511) was erected on a scaffold pole at
an elevation of 5 m above the bed to obtain complimentary measurements to the
ultrasonic distance meters [Blenkinsopp et al., 2010a]. The line scanner was more
highly resolved in the cross-shore than UDMs with an angular resolution of 0.25°,
leading to a cross-shore spatial resolution of 0.025 m near the scanner and 0.2–0.4 m
near the landward and seaward ends of the scaffold frame, respectively. The line
scanner was sampled at 35 Hz. A near-bed pressure sensor (Druck PTX1830) for
measuring forcing conditions was deployed 5 m seaward of the offshore end of the
scaffold frame and sampled at 4 Hz.
Figure 3.5: Images showing the (a) scaffold frame and (b) main instrument bar:
sensors on the main instrument bar are (a) electromagnetic current
meters; (b) Vectrino I velocimeters; (c) Vectrino II acoustic Doppler
current profilers; (d) FOBS; (e) CCPs; the buried PTs are only identified
by their name in the image.
49
Other swash-zone measurements were largely concentrated on a main
instrument bar of the scaffold frame (Figure 3.5a), spanning 2 m in the alongshore
(Figure 3.5b) and located near the mean high water line. Cross-shore and alongshore
velocities (u and v, respectively) were recorded by two Valeport electromagnetic
current meters (EMCMs) positioned nominally at 0.03 and 0.06 m above the bed
(sensors denoted by a in Figure 3.5b). The cross-shore distribution of velocity and
turbulence was recorded with three Nortek Vectrino I acoustic velocimeters (sensors
denoted by b in Figure 3.5b). Each sensor measured the three velocity components (u,
v, and w, where w is the vertical velocity) at nominally 0.05 m above the bed. One
Vectrino I was located on the main instrument bar, with another located 1.5 m seaward
and the other 1.5 m landward. Vectrino I sensors were sampled at 100 Hz. Highly
resolved near-bed velocity profiles (u, v, and w) were recorded with three Vectrino II
profiling velocimeters (PV; sensors denoted by c in Figure 3.5b). The PVs record the
velocity profile at 0.001-m vertical bin spacing over a range of 0.03 m. The lowest PV
was initially deployed so the lower 0.01 m of the profiling range was located within
the bed. The additional PVs were deployed with alongshore offsets of 0.2 m and
vertical offsets of 0.025 m. The PVs arrangement nominally provided the velocity
profile from the bed level to around 0.07 m above the bed at 0.001-m increments. The
PVs were sampled at 100 Hz. The main instrument bar contained a colocated UDM
and also two buried pressure transducers (Figure 3.5b; Druck PTX1830). Buried
pressure transducers were offset by 0.04 m in the vertical and the upper sensor was
initially deployed 0.05 m below the bed. Pressure transducers were used to estimate
infiltration rates and determine water depth, h, after accounting for the sensor distance
below the bed.
50
Figure 3.6: Sideways view of three buried CCPs, indicated by arrows. Downhanging probes are Vectrino-II velocimeters. Yellow tape measure
indicates scale.
Two different sensors were used to measure sediment concentration.
Suspended sediment concentration was recorded using a newly constructed Fiber
Optic Backscatter Sensor (FOBS). This version of the FOBS consisted of two
separate probes (sensors denoted by d in Figure 3.5b). The lower probe contained 10
fiber optic send-receive pairs separated by 0.01 m in the vertical. The lower three
pairs were initially buried to enable data capture under mildly erosive conditions. The
upper 10 pairs had vertical spacing of 0.01, 0.02, 0.03, 0.03, 0.04, 0.05, 0.06, 0.06,
0.06, and 0.07 m. Only channels 1, 2, 4, 6, and 9 were functioning during the BeST
study. The FOBS was sampled at 4 Hz. Sediment concentrations near the bed and in
the sheet flow layer were measured using the Conductivity Concentration Profiler
(Chapter 2). Three CCPs (sensors denoted by e in Figure 3.5b, also Figure 3.6) were
deployed under the main instrument bar with alongshore separation of approximately
51
0.2 m and vertically separated to obtain a larger profile of the sediment concentration
in the sheet flow layer. During deployment, sensors were aligned by eye such that the
electrodes were parallel to the cross-shore direction to reduce wake effects and scour.
CCPs were sampled at 4 Hz.
A 4-m aluminum tower was erected in the dunes landward of the scaffold
frame to mount several cameras (denoted by A in Figure 3.7). A Sony DFW-X710
IEEE1394 protocol (firewire) visible-band, red-green-blue (RGB) camera with a
1024×768 pixel array sampled 5-min image sequences at 5 Hz every 15–30 min
during daylight hours when the swash-zone sensors were actively recording. A FLIR
thermal infrared camera with a 640×480 uncooled microbolometer sensitive in the 7 to 14-μm band sampled 5-min image sequences at 6.25 Hz every 15–30 min when the
swash-zone sensors were actively recording during both daylight and darkness.
Finally, a miniature, downward-looking, visible-band in situ bed camera (IBC) with a
720×576 pixel array was deployed from the instrument bar to identify (1) temporal
phases of the swash cycle when bedload or sheet flow occurred, and (2) the instant
when sediment mobility ceased during the bed settling process when only a thin sheet
of water remained on the beach face at the end stages of backwash. Additionally, the
IBC was used to confirm the correct cross-shore orientation of the CCP sensors and to
confirm that the potential wake effects and scour caused by incorrect alignment were
minimal. The IBC was sampled at 8 Hz.
All sensors were surveyed into the local coordinate system using an electronic
total station. The main instrument bar was located near the mean high water line at a
cross-shore location of x = -68.4 m (black square in Figure 3.3a). The scaffold frame
52
Figure 3.7: Overview of the BeST field location displaying camera tower (A), data
logging cabins (B), instrumentation scaffolding (C).
extended from x = -74.8 m to x = -30.7 m (horizontal extent shown in Figure 3.3a).
All sensors on the scaffold frame and the RGB and thermal cameras were cabled into a
shore-based cabin for power, control, and data acquisition (B in Figure 3.7). The
numerous sensors and cameras were recorded on individual laptop computers.
Recorded data was synchronized in time but measurements were not triggered
simultaneously across different sensors. Time synchronization between sensing
systems was achieved using global positioning system (GPS) time standard
(coordinated universal time, UTC) and network time protocol (NTP) software. One
laptop computer served as the master time server with other laptops receiving the time
code either via wired or wireless connection. Each laptop updated its internal clock
every second with the master time server to avoid clock drift. The elevation above the
53
bed for sensors on the main instrument bar was measured before and following each
tide. Vertical adjustments were made prior to the subsequent tide, as necessary, to
return sensors to their planned elevations.
54
Chapter 4
SHEET FLOW SEDIMENT TRANSPORT DURING
QUASI-STEADY BACKWASH
After a number of preliminary field tests at the Indian River Inlet south beach
(Delaware) and on Wembury beach (UK), the BeST field study was the first formal
deployment of the CCP. This chapter describes results from CCP measurements of
sediment concentration in the sheet flow layer during the BeST study, focusing on the
simplest form of sheet flow observed in the swash zone: quasi-steady backwash where
flow accelerations, pressure gradients and bore turbulence were negligible and the
sheet flow resembled stationary, unidirectional sheet flow. The dissipative beach at
the BeST field site had an infragravity-dominated swash zone, generating many quasisteady backwash events.
4.1
Measurements Quality Control
The first high tide measurement cycles during the BeST study were used to test
the CCP instruments and optimize the sensor positioning. Only measurements from
three high tides (tides 7-9), taken during 13 and 14 October, are discussed in this
chapter. To ensure the quality of the sediment concentration profile measurements
made by the CCP, measurements were discarded if any of the following conservative
criteria were met:
•
Sheet flow occurred with a sheet layer thickness smaller than 5 mm
as this cannot be accurately resolved by the CCP.
55
•
The angle of the flow velocity vector relative to shore normal
exceeded 15°. Flow angle time series were determined using the
lower EMCM velocity measurements. This criterion ensured that
the flow was nearly shore-normal and thus roughly parallel to the
CCP probe. As the probe is 1.6 mm thick and 5.6 mm wide, flow
disturbance and scour will be insignificant when the flow is aligned
with the sensor. Under oblique flow, eddy shedding and scour may
occur near the sensor, altering the sheet flow layer near the sensor.
When the water level was below the lowest current meter, no
velocities and no flow angle could be determined. In this case,
CCP measurements were discarded for lack of a reliable velocity
signal even though it is believed that the CCP still accurately
captures sheet flow sediment concentrations during these instances.
•
The 15 minutes following the first bore arrival during rising tide
were discarded to allow the sand bed to compact and fully saturate
around the CCP probes.
•
When two collocated CCPs both recorded the entire sheet layer
within the profiling window and the calculated sheet thickness
differed by more than 4 mm, the measured sheet thickness was
discarded.
Over the three high tide measurement periods, CCP measurements were made
for a total of 94765 sampling instances (6.5 hours). Sheet flow with a thickness of 5
mm or more was recorded for 25199 sampling instances (112 minutes). 69 % of these
measurement instances occurred under oblique flow or when no current meter data
were available, 3 % occurred during the first 15 minutes of each high tide cycle and
CCP sheet thicknesses disagreed by more than 4 mm for 2 %. As a result, 7416
sampling instances (31 minutes) passed all quality control criteria. Two CCPs
recorded the sheet flow layer simultaneously during many of these instances leaving
10082 sediment concentration profiles that passed the quality control criteria.
The least complex swash-zone sheet flow sediment transport conditions occur
under quasi-steady backwash when effects from bore-generated turbulence, pressure
56
gradients and phase lags are negligible. Quasi-steady backwash conditions were
defined when all of the following criteria were met:
•
The velocity as measured by the lowest EMCM was offshoredirected.
•
Fluid accelerations were small. A threshold for accelerations was
𝑑𝑈
defined as 𝑑𝑡 < 1.2 𝑔 sin 𝛾 = 0.27 𝑚/𝑠 2 , where 𝛾 is the beach
slope, g is gravitational acceleration, and U is the flow velocity
magnitude measured by the lowest EMCM. Swash flow during the
backwash initially accelerates under gravity and the acceleration
decreases at the end of the swash cycle when the gravitational force
is balanced by bottom friction [Hughes and Baldock, 2004].
Acceleration effects are assumed to be small when the flow
accelerates under gravity on a low-sloping beach. The acceleration
threshold used here is an order of magnitude smaller than peak
𝑑𝑈
=
accelerations in oscillatory flow tunnel studies (e.g., � 𝑑𝑡 �
𝑑𝑈
1.33 − 2.00 in O’Donoghue and Wright [2004a]; � 𝑑𝑡 �
𝑚𝑎𝑥
𝑚𝑎𝑥
1.47 − 1.87 in Ruessink et al.[2011]) and corresponds to a
modified Sleath number S [Foster et al., 2006a]
𝑑𝑈
𝑑𝑡
𝑆 = (𝑠−1)𝑔
= 0.017
=
(4.1)
where 𝑠 = 2.65 is the relative density of the sediment. This Sleath
number is an order of magnitude smaller than the limit for pressure
gradient-induced sediment mobilization [Sleath, 1999; Foster et al.,
2006a].
•
Abrupt water depth changes were occasionally recorded by the
pressure transducer during the backwash when no acceleration was
registered. This may either be caused by a small secondary wave
propagating onshore during a backwash, or when a new bore
arrived at the sensor location and initiated a new uprush event while
the near-bed flow velocity was still offshore-directed. These events
𝑑ℎ
were excluded by discarding all measurements when 𝑑𝑡 >
0.05 𝑚/𝑠.
57
•
If the acceleration and water depth thresholds were exceeded for a
particular record, data from 0.75 s before until 0.25 s after the
measurement were also removed from the time series since a
sudden acceleration or bore may stir up sediment that stays
mobilized during later times and to account for small timing
differences between the different instruments.
5313 measurements were taken during backwash events (22 minutes). 22% of
these measurements were rejected due to accelerations and 14% because of water
depth changes. As a result, a total of 3863 measurements were taken during quasisteady backwash (16 minutes). Again, the sheet layer was measured by two CCPs
during some of these instances, resulting in a total of 5365 profiles.
4.2
Example Time Series
A first time series excerpt, collected during tide 9 (October 14, 2011 at 18:52
UTC) is shown in Figure 4.1. A single swash event is shown with a duration of 26 s, a
maximum water depth of 0.11 m (Figure 4.1a), and maximum uprush and backwash
velocities of 1.43 m/s and -1.15 m/s, respectively (Figure 4.1b). Two of the three
CCPs (designated as CCP A and CCP B) deployed during tide 9 were located at
approximately the same vertical elevation and at an alongshore separation of roughly
0.2 m and captured the same sheet flow events. This makes it possible to examine the
repeatability of CCP measurements by comparing measurements by CCPs A and B.
Figure 4.1c shows concentration time series measured by 6 channels separated by 2
mm from CCP A. Channels at -4.9 mm, -2.9 mm and -0.9 mm were below the initial
bed level. The bed eroded by 3.5 mm upon bore arrival (0 s ≤ t ≤ 3 s) resulting in a
reduction of sediment concentration at these elevations as sediment was carried higher
into the water column and landward in the cross-shore direction. Channels at 3.1 mm
and 5.1 mm were initially covered by a thin film of water from a previous swash
58
Figure 4.1: Time series excerpt of swash-zone measurements. (a) Water depth time
series. (b) cross-shore (u, solid line) and alongshore (v, dashed line)
velocity measured 0.03 m above the bed. (c) Time series of measured
concentrations at 6 elevations from CCP A. (d) Contour time series of
sediment concentrations from CCP B. Black line indicates bottom of the
sheet flow layer according to the curve-fitting method described in
section 4.3.1, blue line indicates bottom of sheet flow layer according to
concentration cut-off (c = 0.51, see section 4.3.1), magenta line indicates
top of sheet flow layer (c = 0.08).
event, and upon bore arrival recorded a large sediment concentration. Thus,
concentrations above the initial bed level were in phase with the flow velocities and
concentrations below the bed level were in anti-phase with flow velocities. These
phase relationships correspond to the behavior of the upper sheet flow layer and the
pick-up layer as described by Ribberink and Al-Salem [1995]. Before the arrival of
59
the bore, the channel at 1.1 mm was in the smoothed transition between the sand bed
and water column and therefore few conclusions can be drawn from the time series
from this channel. Between 10 s < t < 17 s, the measured concentration at 1.1 mm
decreased as the general bed level temporarily eroded by roughly 3 mm and
subsequently accreted. A similar behavior for the different channels occurred during
backwash sheet flow, but was obscured by a background bed level change on the order
of 2 mm during the backwash, which shifted the positions of the individual CCP
channels with respect to the bottom. The effect of the bed level change on the
concentration signal of an individual CCP channel was larger than the effect of a timevarying sheet flow layer. For example, the channel at -0.9 mm shows an increase in
sediment concentration at 12.5 s < t < 18 s despite an increase in sheet thickness
because there was concurrent accretion, moving the sensor channel from the sheet
layer (lower concentration) toward the stationary bed (higher concentration).
Sediment concentrations as a function of time and vertical elevation measured by CCP
B are shown in Figure 4.1d. The evolution throughout the swash event is analogous to
Figure 4.1c. Magenta and black lines indicate the top and bottom boundary of the
sheet flow layer as defined in the Section 4.3.1. The blue line indicates an alternative
definition of the bottom of the sheet layer as the contour of c = 0.51 (see also Section
4.3.1).
A second time series excerpt, also collected during tide 9 (October 14, 2011 at
17:30 UTC), is displayed in Figure 4.2. Multiple swash events occurred with gravitytimescale bores superimposed on infragravity-timescale swash cycles with a maximum
depth of 0.22 m. The flow direction alternated multiple times between onshore
(uprush) and offshore (backwash) within a single infragravity swash cycle in
60
conjunction with the gravity-timescale bores. Cross-shore and alongshore velocities
are displayed in Figure 4.2b, with the cross-shore velocity displayed as a thick black
Figure 4.2: Time series excerpt of swash-zone measurements. (a) Water depth. (b)
Cross-shore (solid line) and alongshore (dotted line) velocity. Thick
black line indicates when quasi-steady backwash sheet flow occurred that
complied with all quality-control criteria. (c) Sediment volume fraction
measured by CCP. Magenta (black) line indicates top (bottom) of sheet
layer, yellow vertical lines indicate times when instantaneous profiles are
displayed in panels d-g. (d-g) Instantaneous sediment volume fraction
profiles measured by two collocated CCPs (CCP A and CCP B) during
four instances of the event, indicated by yellow lines in panel c.
61
curve for times when quasi-steady sheet flow occurred that met all quality control
criteria. Sediment concentration measurements (Figure 4.2c) demonstrate that the
sediment bed was active throughout most of the swash event, with the immobile bed
level (black line) nearly continuously changing as a result of local hydrodynamic
forcing and cross-shore and alongshore sediment transport gradients. Black and
magenta lines again indicate the bottom and top boundary of the sheet flow layer as
defined in the section 4.3.1. Smoothing of the sediment concentration profile by the
CCP instrument (see section 2.3.2.3) causes the top and bottom boundary to appear to
be separated by approximately 4-5 mm when the bed is at rest and no sheet flow layer
is present, which was the case during gradual flow reversals (e.g., at 17:30:03 UTC
and 17:30:52 UTC).
When the top and bottom boundaries of the sheet flow layer (black and
magenta lines in Figure 4.2c) are separated by more than 5 mm, sediment is
considered to be mobilized as sheet flow. Sheet flow under quasi-steady backwash
conditions occurred for 5 quality-controlled backwash events during this time
segment, highlighted by thick black lines in Figure 4.2b. Some backwash flows, e.g.,
at 17:30:30-17:30:41 UTC, contained a strong alongshore component, resulting in
flows oblique to the CCP probe and sheet thicknesses were discarded during these
instances. During other instances, such as at 17:31:01-17:31:06 UTC, the backwash
flow was too shallow to be recorded by the lowest current meter, meaning that flow
angle could not be determined. Instantaneous sediment concentration profiles are
displayed in Figure 4.2d-g for 4 backwash sheet flow events indicated by yellow
vertical lines in Figure 4.2c. The agreement between sediment concentration profiles
62
measured by CCP A and CCP B (black solid line and blue dotted line in Figure 4.2dg) indicates repeatability of the measurements.
4.3
4.3.1
Concentration Profile
Top and Bottom Boundary of the Sheet Layer
The top boundary of the sheet flow layer is typically defined as the location
where grains become (on average) separated enough so that intergranular forces
become negligible. Bagnold [1956] defines the top boundary at the volume fraction
where the mean radial separation distance between grains equals one grain diameter,
𝜆 = 1, equivalent to a sediment volume fraction 𝑐 of 0.08. Therefore, 𝑐 = 0.08 was
chosen as the cut-off concentration for the top of the sheet flow layer, similar to
previous sheet flow sediment transport studies (Figure 4.1d and Figure 4.2c, magenta
line) [Dohmen-Janssen and Hanes, 2002; O’Donoghue and Wright, 2004a].
The bottom boundary of the sheet flow layer, the boundary between the sheet
flow layer and the non-moving sediment bed, is more difficult to define since the
concentration of the non-moving sediment bed varies over time under field conditions.
Sediment that is packed at the closest packing limit, around 𝑐 = 0.644 for natural
sands [Bagnold, 1966b], is immobile by definition. Sediment that is packed at a
concentration between the closest packing limit and the loosest packing limit may be
mobile or immobile since it is packed densely enough to support itself statically, but
loosely enough to allow sediment motion. Moreover, the loose packing fraction is
difficult to determine experimentally and is not as well-established as the random
close-packing limit. The loose packing limit is approximately 𝑐 = 0.55 for uniform
spheres [Song et al., 2008], but may be less for natural sands. Bagnold [1966b] found
63
𝑐 = 0.51 for beach sand, and Baker and Kudrolli [2010] found 𝑐 = 0.50-0.54 for
platonic solids which, like natural sand, are angular, lowering the random loose
packing limit. A simple approach to defining the bottom of the sheet flow layer was
applied in a preliminary analysis of the BeST field measurements where the bottom
was defined as the location where the smoothed measured concentration profile
exceeds the loose packing limit, 𝑐 = 0.51 [Lanckriet et al., 2013] (Figure 4.1, blue
line). Although this approach yielded satisfying results for the event displayed in
Figure 4.1, it produced many erroneous estimates for the bottom of the sheet layer in
other locations of the dataset. A more sophisticated method for determining the
bottom of the sheet flow layer is therefore required.
Yu et al. [2012] divided the sheet flow layer into two parts: an upper layer of
‘rapid sediment flow’ with 𝑐𝑙 < 𝑐 < 𝑐𝑡 and a lower layer with 𝑐𝑐 < 𝑐 < 𝑐𝑙 , where
𝑐𝑡 = 0.08, 𝑐𝑙 = 0.57 and 𝑐𝑐 = 0.635 are the concentration at the top of the sheet layer,
the random loose packing concentration and the random close packing concentration,
respectively. In the lower layer, grains are in enduring contact and the flow behaves
like a glassy solid. Experimental results by Capart and Fraccarollo [2011] display a
similar division of the sheet flow layer into two sublayers. Results from a 1DV, twophase numerical model based on this division show that the total sediment transport in
the lower layer is smaller [O(10%)] than in the upper layer for unidirectional,
sinusoidal and skewed oscillatory flow, and that there is a sharp ‘shoulder’ transition
in the concentration profile between the two layers [Amoudry et al., 2008; Yu et al.,
2010]. This shoulder is also observed in measured sheet flow sediment concentration
profiles [O’Donoghue and Wright, 2004a; Dohmen-Janssen and Hanes, 2005]. Since
the shoulder is also observed in sediment concentration profiles measured by the CCP,
64
(a)
0.01
z'(m)
0.005
0
-0.005
-0.01
(b)
0.01
z'(m)
0.005
0
-0.005
-0.01
0
0.1
0.2
0.3
0.4
c (-)
0.5
0.6
0.7
Figure 4.3: Instantaneous sediment concentration profiles recorded during quasisteady backwash events at (a) 17:30:15.75 UTC (Figure 4.2d) and (b)
17:32:1.50 UTC (Figure 4.2g). Black line indicates the measured
concentration profile. Blue line indicates curve fit of the form of
equation 4.3. Red diamond indicates inflection point in curve. Red
upward facing triangle and red downward-facing triangle indicate the top
and bottom of the sheet layer, 𝒛𝒕 and 𝒛𝒆 , respectively.
it was used to define the bottom of the sheet layer using a technique similar to
O’Donoghue and Wright [2004a]. This means that the glassy lower region of the
sheet flow layer, and the small fraction of the total sheet flow transport that may take
place in this layer, was ignored.
65
Instantaneous concentration profiles for two of the backwash sheet flow events
discussed earlier are displayed in Figure 4.3 to illustrate the method to determine the
top and bottom boundaries of the sheet flow layer. To determine the top boundary of
the sheet flow layer, the instantaneous profile (Figure 4.3, black lines) was smoothed
spatially using a boxcar average with a width of 3 mm (3 points in the profile) and the
top of the sheet layer was defined as the elevation where the smoothed concentration
profile equals 0.08 (red upward-facing triangles in Figure 4.3).
The sharp shoulder transition in the sheet flow concentration profiles was
observed in both profiles at volume fractions between 0.51 and 0.55 (Figure 4.3). The
method used to define the bottom boundary of the sheet flow layer is based on
O’Donoghue and Wright [2004a] who fitted a curve to the sheet flow sediment
concentration profile of the form:
𝛽𝛼
𝑐̌ (𝑧) = 𝛽𝛼 +[𝑧−𝑧
1]
(4.2)
𝛼
where 𝑐̌ (𝑧) is the instantaneous sediment concentration profile normalized by the
sediment concentration in the packed bed, 𝑧1 is the first estimate of the bottom of the
sheet flow layer, and 𝛼 and 𝛽 are fitted shape parameters. Since the concentration in
the packed bed varies under field conditions, a curve of the following form is used
instead (referred to hereafter as the ODW curve):
𝛽𝛼
𝑐(𝑧) = 𝑐𝑏 𝛽𝛼+[𝑧−𝑧
1]
𝛼
(4.3)
where 𝑐𝑏 is the concentration in the bed, determined by fitting (4.3) to each of the
instantaneous concentration profiles with 𝛼, 𝛽, 𝑐𝑏 and 𝑧1 as free parameters.
O’Donoghue and Wright [2004a] found that due to the shape of this curve, the
estimated bottom of the sheet layer 𝑧1 is sometimes too low and this is also seen in the
66
curves in Figure 4.3(blue circles). Therefore, they extended a straight line through the
inflection point of the curve (red diamond in Figure 4.3) and defined an improved
estimate of the bottom of the sheet layer, 𝑧𝑒 , as the intersection of the straight line with
the bed concentration 𝑐𝑏 (down-facing triangle). For the profiles displayed in Figure
4.3, the elevation 𝑧𝑒 of the new estimate for the bottom of the sheet layer agrees well
with the location of the shoulder in the profile and 𝑧𝑒 was therefore chosen as the
bottom of the sheet flow layer and is displayed as a black line in Figure 4.1d and
Figure 4.2c. The bottom of the sheet defined by the 𝑐 = 0.51 contour (blue line,
Figure 4.1d) and the curve-fitting method (black line, Figure 4.1d) are in good
agreement for the event shown in Figure 4.1. Throughout the entire dataset, the curvefitting method appeared to be more reliable with less erroneous detections of the bed
level. Defining the bottom of the sheet flow layer by extending the linear portion of
the sheet flow concentration profile is similar to the method used by Pugh and Wilson
[1999]. The sheet flow layer thickness 𝛿𝑠 was defined for each individual profile as
the vertical distance between the top and bottom boundary of the sheet flow layer.
The measured sheet flow layer thickness was then corrected for the smoothing effect
using equation (2.13) (Section 2.3.2.3).
4.3.2
Ensemble-averaged Sediment Concentration Profiles
Sediment concentration profiles under quasi-steady backwash conditions were
grouped according to the measured sheet thickness in 1 mm bins. Individual profiles
were centered around the elevation where the concentration of the smoothed
individual profile (using a 3-point boxcar average) equals 0.30 to align profiles in the
vertical. Other methods, such as centering profiles around the top or bottom of the
sheet layer, or the mean elevation of the top and bottom, resulted in a larger spread of
67
0.025
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
0.02
0.015
z' (m)
0.01
0.005
0
-0.005
-0.01
-0.015
0.025
0.02
0.015
z' (m)
0.01
0.005
0
-0.005
-0.01
-0.015
0.025
0.02
0.015
z' (m)
0.01
0.005
0
-0.005
-0.01
-0.015
0
0.2
0.4
c (-)
0.6
0
0.2
0.4
c (-)
0.6
0
0.2
0.4
c (-)
0.6
Figure 4.4: Ensemble-averaged sediment concentration profiles for sheet thicknesses
of 6 mm (left column), 11 mm (middle column) and 16 mm (right
column). Gray lines indicate individual profiles; dotted black line
indicates ensemble-averaged profile. Horizontal lines indicate top and
bottom of sheet layer. Top row: Orange solid line indicates linear fit.
Middle row: blue solid line and orange dashed line indicate power law
and linear law components of the composite curve fit. Bottom row:
orange solid line indicates ODW curve with 𝛼 = 1.73 fixed.
68
the individual profiles. Using the centered, non-smoothed individual profiles, an
ensemble-averaged profile was calculated for each sheet flow layer thickness. The
average concentration at a particular elevation of the ensemble-averaged profile was
only retained if at least one third of all individual profiles had a valid concentration
measurement at that elevation. Ensemble-averaged profiles for sheet thicknesses of 6
mm, 11 mm and 16 mm are displayed in the left, middle and right columns of Figure
4.4 respectively.
Qualitatively, the bin-averaged profiles appear linear in the lower part of the
sheet flow layer and have a power-law tail toward the top of the sheet flow layer,
particularly for larger sheet thicknesses (Figure 4.4 c,f,i), in agreement with past
studies [Sumer et al., 1996; Pugh and Wilson, 1999]. The measured sediment
concentration in the non-moving sediment bed (z-ze < 0) shows variability of O(0.10).
Laboratory measurements in a packed sediment bed demonstrated that instrument
measurement accuracy is on the order of 0.03 (Figure 2.6). The variability observed in
Figure 4.4 is therefore likely because the upper layer of the non-moving bed was
deposited by natural wave-induced sediment transport and was constantly reworked by
shear, erosion and deposition throughout the high tide measurement cycle. For sheet
thicknesses of 6 mm and 11 mm, the concentration appears to increase monotonically
in the bed. This is likely due to the fact that the sand was initially deposited loosely
and was subsequently packed down by the weight of sediment deposited above. As
with the individual concentration profiles, the top of the sheet layer 𝑧𝑡 was again
determined as the elevation where the ensemble-averaged concentration profile,
smoothed using a 3-point boxcar average, equaled 0.08. The bottom of the sheet layer
𝑧𝑒 and the concentration in the bed 𝑐𝑏 were first determined by fitting a curve of the
69
form (4.3) to the bin-averaged profile and extending the curve from the inflection
point to the bed concentration to determine the bottom of the sheet layer 𝑧𝑒 , as was
done for the individual concentration profiles. This determined the top and bottom of
the sheet flow layer, the sheet flow layer thickness and the concentration in the bed.
Past work has suggested different shapes for the sediment concentration profile
in the sheet layer. Several of these shapes, along with newly proposed shapes, were
compared to the ensemble-averaged measured profiles. First, a linear profile has been
proposed [Hanes and Bowen, 1985; Wilson, 1987]
(4.4)
𝑐(𝑧) = 𝑎1 𝑧 + 𝑎2 ,
where 𝑎1 and 𝑎2 are fitting parameters. Secondly, a power law has also been proposed
as a shape for the concentration profile in the sheet layer [Ribberink and Al-Salem,
1994]
𝑐(𝑧) = 𝑏1 (𝑧′)𝑏2 ,
(4.5)
where 𝑏1 and 𝑏2 are fitting parameters and 𝑧 ′ = 𝑧 − 𝑧𝑒 is the elevation above the
bottom of the sheet layer. Third, Sumer et al. [1996] and Pugh and Wilson [1999]
stated that the profile had a linear shape in the lower section of the layer and a power
law tail at the upper section, defined here as a composite profile:
𝑑1 (𝑧′ − 𝑧𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛 ) + 𝑐𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛
𝑑2
𝑐(𝑧) = �
𝑧′
𝑐𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛 �𝑧
�
𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛
, 𝑧′ ≤ 𝑧𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛
, 𝑧′ > 𝑧𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛
(4.6)
where 𝑑1 and 𝑑2 are fitting parameters, 𝑧𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛 is the elevation above the bottom of
the sheet layer where the profile transitions from a linear to a power law shape and
𝑐𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛 is the concentration at 𝑧𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛 .
70
Fourth, O’Donoghue and Wright [2004a] proposed a curve of the form (4.2).
The ODW curve can be rewritten in terms of the sheet thickness 𝛿𝑠 instead of a
generic length scale 𝛽:
𝑐 = 𝑐𝑏
1
𝑧′ 𝛼 𝑐 −𝑐
1+� � 𝑏 𝑡
𝛿𝑠
𝑐𝑡
(4.7)
,
where 𝑐𝑡 = 0.08 is the concentration at the top of the sheet layer. Each of these curves
was fitted between the top and bottom boundary of the sheet flow layer. For each
shape, the goodness of fit was evaluated using the coefficient of determination 𝑅 2 ,
defined as
∑ (𝑐 −𝑐̅)2
𝑅 2 = 1 − ∑ 𝑖(𝑐 𝑖−𝑐� )2 ,
𝑖
𝑖
(4.8)
𝚤
where 𝑐𝑖 are the discrete concentration values in the ensemble-averaged profile, 𝑐̅ is
the mean of all concentration values 𝑐𝑖 in the ensemble-averaged profile, and 𝑐�𝚤 are the
values predicted by the curve fit. Results for all fitted shapes are summarized in Table
4.1.
Table 4.1:
Sheet thickness
(mm)
# profiles
Summary of fitting results for ensemble-averaged profiles.
6
7
8
9
10
11
12
13
14
15
16
17
18
1574
903
651
418
204
130
97
84
68
64
44
41
42
R : linear
0.97
0.96
0.96
0.97
0.96
0.96
0.94
0.92
0.92
0.90
0.89
0.87
0.87
R2: power law
0.78
0.85
0.90
0.61
0.76
0.83
0.88
0.88
0.69
0.74
0.80
0.81
0.88
R : composite
0.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
𝑐𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛
0.20
0.21
0.22
0.23
0.24
0.24
0.24
0.24
0.25
0.25
0.25
0.29
0.30
0.57
0.54
0.51
0.48
0.46
0.44
0.43
0.39
0.38
0.36
0.35
0.27
0.28
0.98
0.99
0.99
0.99
0.99
0.99
1.00
1.00
1.00
1.00
1.00
1.00
1.00
2.03
1.99
1.94
1.89
1.86
1.82
1.80
1.65
1.61
1.56
1.53
1.44
1.51
0.95
0.97
0.97
0.98
0.99
0.99
1.00
0.99
0.99
0.98
0.98
0.94
0.96
2
2
𝑧𝑡𝑟𝑎𝑛𝑠𝑖𝑡𝑖𝑜𝑛 /𝛿𝑠
2
R : ODW
(𝛼 free)
𝛼: ODW
2
R : ODW
(𝛼 = 1.73)
71
The linear fit (red dotted line, Figure 4.4a,b,c) describes the sheet layer
relatively well for small sheet thicknesses. For larger sheet layers, however, the
discrepancy between the concentration profile and the linear fit increases as the
power-law tail at the top of the sheet layer becomes more prominent. The coefficient
of determination R2 thus decreases from 0.97 for small sheet layer thicknesses to 0.87
for large thicknesses. The power law is not a satisfactory description for the sheet
layer down to the non-moving bed (0.61 ≤ R2 ≤ 0.90) because the concentration
diverges to infinity as 𝑧′ → 0 for a concentration profile that decays away from the bed
(exponent b < 0).
The composite profile (Figure 4.4d,e,f) provides a near-perfect fit for all sheet
layer thicknesses (R2 > 0.99). The transition point between the linear and power-law
segment was chosen so that the overall composite profile had the maximum
coefficient of determination 𝑅 2 , making it possible to determine where the profile
transitions from a linear to a power-law shape. The concentration profile transitions to
a power-law shape at concentrations in the range 0.20 – 0.30 and at elevations of 0.27
𝛿𝑠 - 0.57 𝛿𝑠 (Table 4.1). The results are not sensitive to the location of the transition
point so the elevation of the transition point (and the associated concentration) can be
changed by several millimeters without significantly reducing the goodness of fit.
The ODW curve expressed as equation (4.7) was fitted to the data with the
sheet thickness 𝛿𝑠 and the concentrations at the top and bottom of the sheet layer, 𝑐𝑏
and 𝑐𝑡 , all held fixed, and with the shape parameter 𝛼 as the only free parameter,
resulting in an excellent fit to the ensemble-averaged concentration profiles (𝑅 2 ≥
0.98). The shape parameter 𝛼 varied between 1.44 and 2.03, in good agreement with
the range of values for 𝛼 = 1.1-1.9 found by O’Donoghue and Wright [2004a] for
72
Figure 4.5: Individual sheet flow sediment concentration profiles normalized by
sheet thickness. Black line indicates ensemble average of all normalized
profiles, white dotted line indicates ODW curve with 𝛼 = 1.73.
time-averaged profiles in oscillatory flows. The small range of values for 𝛼 indicates
a high level of similarity between ensemble-averaged sediment concentration profiles
of varying thickness. Therefore, a second ODW curve was compared to the data with
𝛼 held constant at 1.73, the mean value across all sheet thicknesses. This curve again
agreed well with the ensemble-averaged profiles (R2 ≥ 0.94; red dotted line in Figure
4.4g,h,i).
The ODW shape is similar to the composite linear-power law profile because
the ODW curve has an inflection point, making the curve locally linear in the lower
section of the sediment concentration profile, and it reduces to a power law for 𝑧→∞.
The only difference with the composite profile is that the transition between linear and
power-law behavior occurs smoothly in the case of the ODW curve. The finding that
73
𝛼 varies relatively little in the ODW curves with 𝛼 as a fitting parameter indicates that
there is a high level of self-similarity between sheet flow sediment concentration
profiles of varying sheet thickness. This self-similarity is demonstrated in Figure 4.5,
in which all 5365 individual profiles are displayed, scaled by their sheet thickness so
that the vertical coordinates equal 0 and 1 at the bottom and top of the sheet flow
layer, respectively. The solid black curve indicates the ensemble average of all
normalized profiles.
4.4
Sheet Thickness and Sheet Load
Hydrodynamic forcing for unidirectional sheet flow is typically described
using the Shields number 𝜃:
or the mobility number 𝜓:
𝜃=𝜌
𝜏𝑏
(4.9)
𝑈2
(4.10)
𝑓 (𝑠−1)𝑔𝑑
𝜓 = (𝑠−1)𝑔𝑑 ,
where 𝜏𝑏 is the bed shear stress, 𝜌𝑓 is the fluid density and 𝑑 is a representative grain
diameter which is typically defined as the median grain diameter 𝑑50 . Wilson [1987]
proposed a linear relationship between the sheet thickness 𝛿𝑠 and the Shields number:
𝛿𝑠
𝑑
= Λ𝜃.
(4.11)
For stationary flow, Wilson [1987] proposed Λ = 10 whereas Sumer et al. [1996]
found Λ = 11.8. Different values for the peak sheet layer thickness under oscillatory
flow were proposed ranging from 13 to 35. [e.g., Dohmen-Janssen et al., 2001; Dong
et al., 2013].
The bed shear stress 𝜏𝑏 is commonly expressed by a quadratic drag law:
74
1
𝜏𝑏 = 2 𝑓𝜌𝑓 𝑈 2 ,
(4.12)
where 𝑓 is a friction factor. The bed shear stress is difficult to determine directly in
the swash zone and large uncertainties exist on the value of the friction factor
[Hughes, 1995; Puleo and Holland, 2001; Conley and Griffin Jr, 2004; Raubenheimer
et al., 2004; Puleo et al., 2012]. Since the mobility number does not depend on the
friction factor, it was chosen to represent the hydrodynamic forcing for this section.
The correlation was calculated between mobilized sediment, represented by
sheet thickness and sheet load (equation 2.16), and hydrodynamic forcing, represented
by the mobility number 𝜓. Sheet thickness was scaled by 𝑑50 , similar to previous
predictions of sheet thickness as a function of hydrodynamic forcing (equation 4.11),
and sheet load was scaled by 𝑑50 𝜌𝑠 . During sampling instances when more than one
0.03
18
(a)
(b)
16
0.025
14
12
2
C (kg/m )
δs (m)
0.02
0.015
10
8
6
0.01
4
0.005
2
0
0
200
400
ψ (-)
600
0
800
0
200
400
ψ (-)
600
800
Figure 4.6: (a) Sheet flow thickness 𝛿𝑠 as a function of mobility number 𝜓, (b) Sheet
load 𝐶 as a function of mobility number 𝜓. Solid line indicates best fit
through origin; dashed lines indicate factor two range. No measurements
are displayed with sheet layer thicknesses less than 5 mm as they cannot
be resolved by the CCP.
75
CCP captured the sheet flow layer, the mean of the sheet thicknesses and sheet load
measurements from both instruments was used so that a single sampling instance was
only counted once in the correlation. Scatter plots displayed in Figure 4.6 show the
sheet thickness (a) and sheet load (b) as a function of the mobility number.
The square of the Pearson correlation coefficient 𝑟 2 between sheet thickness
and mobility number is 0.60, and the correlation coefficient between sheet load and
mobility number is 0.53. The relationship between sheet thickness or sheet load and
the mobility number was also assessed using a linear fit through the origin (solid line)
similar to the linear relation between sheet thickness and Shields number (equation
4.11) proposed by Wilson [1987]. The resulting linear fits were:
𝛿𝑠
𝑑50
𝐶
𝑑50 𝜌𝑠
= 0.108 𝜓
= 0.0272 𝜓 .
(4.13)
(4.14)
The root mean square errors of the linear fits through the origin were 10.2 for the sheet
thickness and 2.81 for the sheet load. 73% of all profile measurements were within a
factor of two of the fit through the origin (dashed lines) for the sheet thickness and
69% of all profile measurements for the sheet load. The agreements between sheet
thickness and mobility number and sheet load and mobility number were similar,
because the sheet thickness determined the integration limits for the calculation of the
sheet load, and thus calculated sheet thickness and sheet load were highly correlated
(𝑟 2 = 0.96).
Figure 4.7 reprises the same time series excerpt as in Figure 4.2 and shows
time series of the measured mobility number, sheet thickness and sheet load. Sheet
layer thickness exceeded 5 mm for five backwash events. Sheet thickness and sheet
76
(a)
h (m)
0.2
0.1
0
(b)
u, v (m/s)
1
0
-1
(c)
ψ (-)
300
200
100
0
(d)
δs (m)
0.02
0.01
Sheet load (kg/m2 )
0
(e)
10
5
0
17:30:00
17:30:30
17:31:00
Time (UTC)
17:31:30
17:32:00
Figure 4.7: Time series excerpt of swash zone measurements. (a) Water depth. (b)
Cross-shore (solid line) and alongshore (dashed line) velocity. (c)
Mobility number. Thick black lines in panels b and c indicate times
when quasi-steady backwash sheet flow occurred. (d) Sheet thickness
and (e) sheet load. Black dots indicate measurements during quasisteady backwash, gray dots indicate measurements during uprush sheet
flow events that met quality control criteria and during backwash sheet
flow events when no current meter data was available. Orange crosses
indicate predictions by fits through origin (equations 4.18-19).
77
load typically increased monotonically during the backwash as the backwash velocity
increased. Measurements at the end of backwash events when the flow depths became
smaller than the elevation of the current meter indicate that the sheet thickness
decreased at the end of the backwash, when flow velocities also decreased as bottom
friction became dominant (e.g., at 17:31:01-17:31:07 UTC). In addition, uprush sheet
flow events were also observed, e.g., at 17:30:47 UTC.
Sheet thickness and sheet load were also predicted by the linear fits (4.13-14).
These values (orange crosses in Figure 4.7d-e) show that a prediction based on the
mobility number alone reproduced the trend of increasing sheet thickness with
increasing shear stress. The linear models generally underpredicted measured sheet
thickness and sheet load in the time excerpt shown. This is due to the fact that the
linear fits were prescribed to intersect with the origin, and because sheet flows with a
thickness less than 5 mm are not resolved by the CCP, as seen in the scatter plots in
Figure 4.6. Most measurements with 𝛿𝑠 ≤ 11 𝑚𝑚 were underpredicted by the model
(Figure 4.6a), as sheet thickness measurements that were overpredicted would likely
not be resolved by the CCP. Still, the linear fit through the origin provides the most
adequate model for sheet thickness and sheet load because sheet flow layer thickness
is zero when no hydrodynamic forcing is present and because it is supported by
previous literature [Wilson, 1987; Sumer et al., 1996]. With root mean square errors
of 8.93 for the sheet thickness and 3.14 for the sheet load for the time series excerpt
displayed in Figure 4.7, the agreement with the linear fit is representative for the entire
dataset.
78
4.5
Discussion
The dataset presented here consists of 5365 individual sediment concentration
profile measurements from a total of 423 sheet flow events, taken during three high
tide cycles. Quality-controlled sheet flow with a sheet layer thickness exceeding 5
mm was recorded during 32 minutes out of 6.5 hours of measurements, meaning that a
significant sheet flow layer was present for at least 8 % of the time and probably more,
since sheet flow almost certainly also occurred during oblique flows and flows with a
free surface level below the current meter that did not meet quality control criteria.
This result demonstrates that significant sheet flow frequently occurs in the swash
zone.
Backwash flows were deemed quasi-stationary when the measured flow
acceleration was below a threshold of 1.2 𝑔 sin 𝛾 = 0.27 𝑚/𝑠 2 . The analysis was
repeated with an acceleration threshold reduced by a factor of four to 0.3 𝑔 sin 𝛾 =
0.07 𝑚/𝑠 2 to verify that flow accelerations below this threshold still had a negligible
effect on the sheet flow. This lower threshold reduced the number of measurements
that passed quality-control criteria from 3863 to 413. Fitting coefficients in the linear
models for sheet thickness and sheet load (equation 4.13-4.14) changed by less than
2%, while the rmse errors for the fits decreased by 5% and 6% respectively. The
small variations demonstrate that flow acceleration effects were indeed negligible for
the original threshold of 0.27 𝑚/𝑠 2 .
Backwash sheet flow when the water level was below the elevation of the
lowest current meter could have been included in the analysis by estimating the depthaveraged velocity using a continuity technique [Blenkinsopp et al., 2010a, 2010b]. A
lidar scanner and an array of ultrasonic distance meters (UDMs) that are capable of
generating an estimate of the depth-averaged velocity were deployed during the field
79
study. However, only velocity measurements by the electromagnetic current meters
were used in this chapter because combining velocity measurements from different
instruments could provide an additional source of variability in the dataset, and this
may obfuscate the relationship between the sheet flow layer thickness and
hydrodynamic forcing.
Besides during the backwash, sheet flow was also observed during the uprush,
e.g., at 17:30:46 - 17:30:48 UTC in Figure 4.7. Sheet flow during uprush is more
difficult to analyze than quasi-steady backwash sheet flow since it is influenced by
additional forcing mechanisms such as bore-generated turbulence, pressure gradients
and phase lags. For this reason, this chapter focused on sheet flow during quasi-steady
backwash conditions.
The sheet flow sediment concentration profile had a linear shape in the lower
section of the sheet layer and a power law shape in the upper section of the sheet layer,
with the transition between the two shapes occurring at sediment concentrations of
0.20-0.30. The profile was also self-similar for different sheet layer thicknesses. A
linear shape, assumed by some sheet flow sediment transport models such as Hanes
and Bowen [1985] and Wilson [1987], was in poor agreement with measured
concentration profiles. A curve that incorporates the linear and power law, such as a
composite profile or the empirical curve by O’Donoghue and Wright [2004a], was in
excellent agreement with the field observations.
Adopting a curve such as the combined linear and power law or the empirical
curve by O’Donoghue and Wright [2004a] can lead to improved sheet flow sediment
transport models. For example, Wilson [1987] provided an analytical derivation of the
sheet thickness by assuming that the bed shear stress 𝜏𝑏 at the bottom of the sheet
80
layer is balanced by an intergranular shear stress, expressed as a Coulomb failure
criterion:
𝜏𝑏 = 𝜎𝑠 tan 𝜙 ′ ,
(4.15)
𝑧
(4.16)
where 𝜙 ′ is the dynamic friction angle of the solids and 𝜎𝑠 is the intergranular normal
stress:
𝜎𝑠 = ∫𝑧 𝑡 𝜌𝑓 𝑔(𝑠 − 1)𝑐(𝑧)𝑑𝑧 .
𝑒
Wilson [1987] assumed a linear concentration profile 𝑐(𝑧) in the sheet layer and a
sediment concentration of zero at the top of the sheet layer, yielding:
𝜎𝑠 = 𝜌𝑓 𝑔(𝑠 − 1)
Combining (4.15) and (4.17) yields
𝛿𝑠
𝑑
=𝑐
𝑏
2
tan 𝜙′
𝜏𝑏
𝜌𝑓 𝑔(𝑠−1)𝑑
=𝑐
𝛿𝑠
2
(4.17)
𝑐𝑏 .
2
𝑏 tan 𝜙
′
𝜃 = Λ𝜃 .
(4.18)
Wilson [1987] assumed 𝑐𝑏 = 0.625 and tan 𝜙 ′ = 0.32, leading to Λ = 10. Assuming
a linear law with a power-law tail at the top of the sheet flow layer instead leads to
higher values for Λ. For example, evaluating 𝜎𝑠 using the ODW curve with 𝛼 =1.73
(equation 4.7) and assuming 𝑐𝑡 = 0.08 at the top of the sheet layer leads to Λ = 11.8,
the same value as found by Sumer et al. [1996]. The exact value of Λ is also
influenced by assumptions on the sediment concentration in the bed 𝑐𝑏 and the
dynamic friction angle 𝜙′, but the improved description of the shape of the
concentration profile nevertheless leads to sheet thickness estimates that are in better
agreement with previous measurements.
Sheet thickness has previously been predicted as a linear function of the
Shields number (equation 4.11). Because of the difficulty in determining the bottom
shear stress in the swash zone, sheet thicknesses and sheet load were related to the
81
mobility number 𝜓 in this study. The mobility number 𝜓 can be related to the Shields
number 𝜃 when a constant friction factor is assumed:
1
𝜃 = 2 𝑓𝜓 ,
(4.19)
By using equation (4.19), the linear fit between mobility number and the sheet
flow layer thickness based on the dataset analyzed in this study is equivalent to the
linear law for steady flow based on the Shields number (equation 4.11) with Λ = 11.8
[Sumer et al., 1996] when a friction factor 𝑓 = 0.018 is assumed. Velocity profile
measurements in the bottom boundary layer during high tide 7, assuming a logarithmic
velocity profile, yielded 𝑓 = 0.021 ± 0.012 during backwash flows [Puleo et al.,
2014b]. The small difference between the friction factor estimates indicates that the
sheet thickness measurements found here are in good agreement with past predictions
based on the Shields number. This result, however, does not contravene the fact that
there is still significant scatter associated with field measurements of bed shear stress,
friction factors and sheet flow layer thicknesses in the swash zone, as illustrated by the
scatter in Figure 4.6 for sheet layer thickness and the standard deviation in the friction
factor estimate based on velocity measurements. Additionally, sheet thickness values
may vary between different studies because they were based on different measurement
techniques, e.g. concentration measurements [e.g. Sumer et al., 1996; DohmenJanssen and Hanes, 2005, this study] or visual methods [e.g. Dong et al., 2013] or
because they were based on different definitions for the top and bottom boundary of
the sheet flow layer.
Laboratory measurements have shown that sheet flow initiates above a
threshold Shields number of 0.5 – 1.0 [Nielsen, 1992; Asano, 1995] or a threshold
mobility number of approximately 240 [Dingler and Inman, 1976]. Including a
82
threshold value in the linear models (equations 4.13 - 4.14), similar to the threshold
value in the bedload transport formula by Meyer-Peter and Müller [1948], could
therefore yield a more accurate representation of the relationship between sheet
thickness, sheet load and the mobility number. A threshold value was not included in
the linear models (equations 4.13 - 4.14) to facilitate comparison with existing
relationships between sheet thickness and bed shear stress that also do not include a
threshold value, such as the relationships proposed by Wilson [1987] and Sumer et al.
[1996]. In addition, accurately resolving a threshold value would require
measurements of small sheet thicknesses near the threshold but these were not
resolved by the sensor.
Sheet load magnitudes of up to 14 kg/m2 were frequently observed during
quasi-steady backwash events. Suspended loads of up to 4 kg/m2 were observed in a
preliminary analysis of fiber-optic backscatter sensor (FOBS) measurements during
high tide 7 and were typically O(1-10) times smaller than concurrent sheet load
magnitudes [Puleo et al., 2014b], indicating that near-bed sediment loads are not
negligible in the swash zone.
4.6
Conclusions
Sediment concentration profiles in the sheet flow layer were measured in the
swash zone of a dissipative beach using three conductivity concentration profilers,
providing the first detailed measurements of sheet flow under field conditions. This
chapter focused on sheet flow during quasi-steady backwash conditions when the
effects of surface-generated turbulence, phase lags and pressure gradients on the sheet
flow layer were negligible, simplifying the hydrodynamic forcing for the sheet flow.
From 6.5 hours of measurements spanning three high tides, 16 minutes of sheet flow
83
was recorded during quality controlled, quasi-steady backwash conditions, providing a
dataset of 5365 instantaneous sediment concentration profiles.
The concentration profiles were grouped according to sheet flow layer
thickness and ensemble-averaged. The sheet flow sediment concentration profile had
a linear shape in the lower section and a power-law shape in the upper section, with
the transition between the two shapes occurring at sediment volume fractions of 0.200.30. The profile shape was self-similar for sediment concentration profiles of varying
sheet thicknesses ranging from 6 mm to 18 mm and can be described by a curve based
on O’Donoghue and Wright [2004a]. Adopting this curve as the sediment
concentration profile provides an improvement to the analytical model for sheet flow
layer thickness developed by Wilson [1987]. This linear-power law curve shape also
has significance toward future models of nearbed sediment transport where the
sediment concentration profile in the sheet flow layer requires parameterization.
The sheet flow layer thickness and sheet load were well-correlated to the
hydrodynamic forcing represented by the mobility number 𝜓, with r2 = 0.60 for the
sheet layer thickness and r2 = 0.53 for the sheet load. A simple model with the
nondimensional sheet thickness and sheet load proportional to the mobility number
provided a good prediction of a time series of sheet thickness and sheet load.
84
Chapter 5
TURBULENCE DISSIPATION
Turbulence is a key factor in swash-zone hydrodynamics and sediment
transport: the balance between uprush (onshore-directed) and backwash (offshoredirected) transport may not be explained without accounting for bore turbulence
[Puleo et al., 2000; Butt et al., 2004; Aagaard and Hughes, 2006]. Highly-resolved
velocity profile measurements were collected during the BeST field study (Chapter 3).
The purpose of this chapter is to estimate the turbulence dissipation rate based on
these velocity measurements using the structure function, in order to investigate 1)
temporal variability of the turbulence dissipation rate and its relationship to imagebased observations of wave breaking, 2) the decay of turbulence dissipation following
bores, 3) the structure of the vertical dissipation profile and its implications toward
turbulence generation mechanisms, 4) the dependence of the turbulence dissipation
rate on forcing parameters such as (tidally modulated) local water depth and offshore
wave height and 5) the relative importance of turbulence dissipation and density
stratification in the TKE budget.
5.1
Introduction
Few field measurements of the turbulence dissipation rate 𝜖 have been
performed in the swash zone because of measurement difficulties [Flick and George,
1990; Rodriguez et al., 1999; Raubenheimer et al., 2004] and most of the present
knowledge on swash zone turbulence stems from laboratory [e.g. Petti and Longo,
85
2001; Cowen et al., 2003; O’Donoghue et al., 2010; Sou et al., 2010] and numerical
[e.g. Zhang and Liu, 2008; Torres-Freyermuth et al., 2013] studies. More field
measurements of the dissipation rate are available in the surf zone, where velocimeters
are continuously submerged. Measured surf zone dissipation rates may provide an
indication of the expected range of dissipation rates in the swash zone. Dissipation
rates obtained from field experiments in the surf and swash zones ranged from
𝑂(10−7 𝑚2 /𝑠 3 ) to 𝑂(10−1 𝑚2 /𝑠 3 ) (Table 5.1). Measured dissipation rates follow a
lognormal distribution [George et al., 1994] and can vary by up to 3 orders of
magnitude within a single field study, highlighting the variability of turbulence
dissipation.
Turbulence in the swash zone is generated locally by bores (broken waves) and
by bed shear. Turbulence generated by breaking waves in the surf zone is also
advected into the swash zone, meaning that swash-zone turbulence dissipation exceeds
local production and the swash zone acts as a sink for surf-zone turbulence [Sou et al.,
2010]. The relative importance of breaking-wave and bed-generated turbulence can
be assessed from the vertical dissipation profile. Seaward of the surf zone, Feddersen
et al. [2007] observed that the dissipation profile had a maximum near the surface (due
Table 5.1:
Surf and swash zone dissipation estimates.
Source
Flick and George [1990]
George et al. [1994]
Veron and Melville [1999]
Trowbridge and Elgar [2001]
Bryan et al. [2003]
Grasso et al. [2012]
Feddersen et al. [2012a]
Feddersen et al.[2012b]
Raubenheimer et al. [2004]
𝜖(𝑚2 /𝑠 3 )
−2
−2
1 ⋅ 10 − 4 ⋅ 10
5 ⋅ 10−5 − 5 ⋅ 10−2
1 ⋅ 10−3
1⋅ 10−5 − 2 ⋅ 10−4
3⋅ 10−7 − 3 ⋅ 10−3
2⋅ 10−4 − 6 ⋅ 10−3
1⋅ 10−4 − 1 ⋅ 10−3
1⋅ 10−4 − 2 ⋅ 10−3
4⋅ 10−2 − 1 ⋅ 10−1
𝐻𝑠 (m)
Unspecified
0.50-1.20 (offshore)
0.7-1.8
(offshore)
0.50-2.6 (offshore)
0.10-0.62 (local)
1.0-8.0
(offshore)
0.02 - 0.95 (local)
0.5 - 1.7 (local)
0.55-1.0 (offshore)
86
Bottom slope
Unspecified
0.025
0.03
0.01
0.01-0.06
0.025
0.01
0.02-0.05
0.02
Location
Surf
Surf
Surf
Surf
Surf
Surf
Surf
Surf
Swash
to whitecapping) and a secondary maximum near the bed (due to bed shear). In the
surf zone, dissipation is largest near the surface, indicating that wave breaking is
dominant [George et al., 1994; Huang et al., 2009; Grasso et al., 2012]. In the swash
zone, bed-generated turbulence is dominant during the backwash phase. During the
uprush phase, (advected) surface-generated turbulence is dominant, though bedgenerated turbulence may still be significant [Petti and Longo, 2001; Sou et al., 2010].
As expected, bed-generated turbulence is more pronounced in swash experiments over
rough beds than over smooth beds [O’Donoghue et al., 2010].
Bryan et al. [2003] and Raubenheimer et al. [2004] observed that turbulence
dissipation decreases with increasing mean water depth in the surf and the swash
zones. Bryan et al. [2003] also found that ϵ is negatively correlated to ℎ𝛾/𝐻𝑠 where h
is the mean water depth, 𝐻𝑠 is significant wave height and γ is an empirical wave
breaking parameter, 𝛾 = (𝐻𝑠 /ℎ)𝑏𝑟𝑒𝑎𝑘𝑖𝑛𝑔 . Conversely, Grasso et al. [2012] did not
detect a significant depth dependence for ϵ in the surf zone, but did find ϵ to be
dependent on wave skewness, asymmetry, and on 𝐻𝑡𝑜𝑡 /ℎ, where 𝐻𝑡𝑜𝑡 is the total
significant wave height including infragravity waves. Cowen et al. [2003] showed
that for both spilling and plunging monochromatic breakers, Eulerian measurements of
turbulence dissipation decay rapidly after bore arrival following a 𝑡 −2.3 decay rate, the
same rate as freely decaying isotropic grid turbulence. In the case of grid turbulence,
the turbulent kinetic energy (TKE) decays as 𝑡 −1.3 and 𝜖 decays as 𝑡 −2.3 since the
turbulent kinetic energy budget reduces to 𝜖 = −𝑑(𝑇𝐾𝐸)
[Pope, 2000]. In a numerical
𝑑𝑡
study of swash flows, Zhang and Liu [2008] confirmed that the 𝑇𝐾𝐸 decays as 𝑡 −1.3
for strong bores, but that it decays slower, as 𝑡 −0.65, for weak bores. For deep-water
breaking waves, Rapp and Melville [1990] estimated a dissipation decay rate as 𝑡 −2.5
87
whereas Veron and Melville [1999] observed a rate of 𝑡 −𝑛 with 𝑛 ranging between 1
and 1.25.
5.2
5.2.1
Methodology
Existing Analysis Methods
The turbulence dissipation rate can be calculated directly from the fluctuating
strain rate tensor when the velocity field is quantified in two dimensions, e.g. using
Particle Image Velocimetry (PIV) in the laboratory [Cowen et al., 2003; Sou et al.,
2010]. Other methods, such as those based on the wavenumber spectrum, the
structure function and the frequency spectrum employ the Kolmogorov hypotheses to
estimate the dissipation rate from the structure of the velocity field. Veron and
Melville [1999] collected one-dimensional velocity profiles using a pulse-to-pulse
coherent acoustic Doppler velocity profiler (ADVP) and estimated the dissipation rate
based on the one-dimensional wavenumber spectrum, which takes the form
𝐸(𝑘) = 𝛼𝜖 2/3 𝑘 −5/3 ,
(5.1)
in the inertial subrange, where 𝑘 is the wavenumber and 𝛼 = 1.5 is an empirical
coefficient [Pope, 2000]. Another approach based on one-dimensional velocity
profiles is the structure function [Kolmogorov, 1991]. For the vertical velocity
component 𝑤, the second-order longitudinal structure function D is defined as
2
𝐷(𝑧, 𝑟, 𝑡) = ⟨�𝑤 ′ (𝑧 + 𝑟, 𝑡) − 𝑤 ′ (𝑧, 𝑡)� ⟩
(5.2)
where r is the separation distance, t is time and ⟨… ⟩ denotes time averaging. The
structure function takes the form
𝐷(𝑧, 𝑟, 𝑡) = 𝐶 𝜖(𝑧, 𝑡)2/3 𝑟 2/3
88
(5.3)
in the inertial subrange, where 𝐶 = 2.0 is an empirical constant [Pope, 2000]. The
structure function has thus far only been applied to oceanic environments in water
depths greater than several meters such as tidal channels, river mouths and dense
bottom plumes [Wiles et al., 2006; Mohrholz et al., 2008; Whipple and Luettich,
2009].
Many field studies of nearshore hydrodynamics collect only one or a small
number of point measurements of the velocity field and employ Taylor’s frozen
turbulence hypothesis to obtain a dissipation estimate from the (temporal) frequency
spectrum. For a stationary current,
24
𝑃𝑤𝑤 (𝑓) = 55 (2𝜋)−2/3 𝛼𝜖 2/3 𝑉 2/3 𝑓 −5/3
(5.4)
in the inertial subrange, where 𝑃𝑤𝑤 (𝑓) is the frequency spectral density of the vertical
velocity, 𝑓 is frequency, and 𝑉 is the advection velocity (similar expressions exist for
the other velocity components). Lumley and Terray [1983] extended Taylor’s
hypothesis to the case of nearshore turbulence advected by wave orbital motions and a
stationary current [extended further by Trowbridge and Elgar, 2001; Bryan et al.,
2003; Feddersen et al., 2007; Gerbi et al., 2009]. Estimating turbulence dissipation
based on frequency spectra has a number of limitations compared to estimates based
on spatial velocity measurements. For example, Veron and Melville [1999] found that
the dissipation rate calculated using the frequency spectrum and Taylor’s hypothesis
was overestimated by a factor of 4 compared to the dissipation rate calculated from the
same dataset using the wavenumber spectrum. Secondly, frequency spectra are
typically estimated using velocity time series with a duration on the order of minutes,
precluding analysis of the temporal variability of 𝜖 on a wave time scale [O(10 s)].
Velocity time series in the swash zone are discontinuous due to the intermittent flow
89
coverage, meaning that a frequency spectrum based on a long-duration time series
cannot be readily generated. Moreover, swash flows on dissipative beaches are often
dominated by infragravity motion, making it difficult to separate mean and orbital
flow velocities. George et al. [1994] avoided this problem by computing frequency
spectra over short time intervals with a duration of 0.125 s. Dissipation estimates by
George et al. [1994] may have been overestimated because they did not account for
the bias introduced by using a small number of points in the spectral estimate [Bryan
et al., 2003].
5.2.2
Laboratory Validation
The wavenumber spectrum, frequency spectrum and structure function are
illustrated and compared using an idealized laboratory test case under a stationary
flow. The test was conducted in a recirculating flume with a test section width of 0.40
m and depth of 0.46 m. A single layer of rock with a diameter of approximately 0.04
m was placed over a length of 1.6 m of the flume bottom to increase bottom roughness
and enhance turbulence production. Velocity profiles were collected by a pulsecoherent acoustic Doppler profiling velocimeter of the same type as used in the BeST
field study [Nortek Vectrino-II Profiler; Craig et al., 2011]. The velocimeter
measured a vertical profile of all three velocity components (u, v, w) over a range of
0.030 m with a profile resolution of 0.001 m (31 points) using a central acoustic
emitting transducer and four receiving transducers, and was deployed with the velocity
profiling section located 0.12 m – 0.15 m above the flume bottom (~ 0.08 m – 0.11 m
above the roughness elements). A velocity record of 1393 s (23.2 minutes) was
collected at a sampling rate of 100 Hz. Mean streamwise flow velocities varied from
90
Table 5.2:
Turbulence dissipation rate estimates for stationary flow laboratory test.
𝜖 (10−4 𝑚2 𝑠 −3 )
𝑤1
𝑤2
Structure
function
Wavenumber
spectrum
Frequency
spectrum
2.04
2.10
2.07
1.85
1.95
1.77
Frequency
spectrum
(95% high)
2.74
2.49
Frequency
spectrum
(95% low)
1.44
1.31
0.62 – 0.68 m/s across the sampled section of the boundary layer. Resulting
turbulence dissipation estimates are summarized in Table 5.2.
For similar acoustic velocimeters with three receiving transducers in a
downward-facing setup, noise levels of the measured horizontal velocity components
are approximately 31 times higher than the vertical velocity component due to the
transformation of the along-beam velocity components to the velocity components in a
Cartesian coordinate system [Voulgaris and Trowbridge, 1998]. Based on the velocity
transformation matrix for a Vectrino-II probe geometry with four receiving
transducers, the ratio of horizontal velocity noise levels to vertical velocity noise
levels ranges from 10 to 16 along the measured velocity profile. Having lower noise
levels, vertical velocity measurements were used for turbulence estimates in this study
[see also Huang et al., 2012]. Two independent estimates of the vertical velocity, 𝑤1
and 𝑤2 , were derived from the four receiving transducers: 𝑤1 was estimated from
transducers located on prongs of the velocimeter that were oriented perpendicular to
the flow and 𝑤2 from transducers on prongs that were aligned with the flow. The
difference between 𝑤1 and 𝑤2 was used to assess the instrument accuracy.
To calculate the dissipation rate based on the structure function, vertical
velocity fluctuations 𝑤′ were computed by subtracting the time-averaged vertical
velocity from the vertical velocity time series
𝑤 ′ (𝑧, 𝑡) = 𝑤(𝑧, 𝑡) − ⟨𝑤(𝑧, 𝑡)⟩
91
(5.5)
For each elevation bin 𝑧𝑖 in the velocity profile and each separation distance 𝑟𝑗 , the
velocity differences in the structure function were implemented using the central
differences method:
2
𝐷�𝑧𝑖 , 𝑟𝑗 , 𝑡� = ⟨�𝑤′�𝑧𝑖 + 𝑟𝑗 ⁄2 , 𝑡� − 𝑤′�𝑧𝑖 − 𝑟𝑗 ⁄2 , 𝑡� � ⟩ .
𝐷�𝑧𝑖 , 𝑟𝑗 , 𝑡� was then fitted to
2
3
𝐷�𝑧𝑖 , 𝑟𝑗 , 𝑡� = 𝑁 + 𝐴𝑟𝑗 .
(5.6)
(5.7)
where 𝑁 and 𝐴 are fitting parameters. A is indicative of the rate at which the velocity
field decorrelates with separation distance due to turbulence and N is indicative of
measurement (Doppler) noise [Wiles et al., 2006]. Following Wiles et al. [2006], 𝜖
was obtained from equation (5.7) as
𝐴 3/2
𝜖 = �𝐶 �
.
(5.8)
For the laboratory test, the structure function 𝐷(𝑧, 𝑟) was calculated for the
center bin of the vertical velocity profile and the time averages in equations (5.5-6)
were calculated over the entire 23.2-minute velocity record (Figure 5.1a). The
structure function followed an 𝑟 2/3 law for separation distances 𝑟 up to 0.023 m
(𝑅 2 = 0.96); velocity records for larger separation distances were rejected by
velocimeter quality control criteria (Section 5.3). Deviations from the 𝑟 2/3 law were
larger for 𝑤2 than for 𝑤1.
The wavenumber spectrum 𝐸(𝑘) was calculated for the top 30 bins in the
velocity profile (since an even number of points was needed for the Fourier transform)
for each of the 139264 instantaneous profiles. Individual profiles were detrended but
not tapered or band averaged. Instantaneous wavenumber spectra were then timeaveraged to obtain a smooth wavenumber spectrum (Figure 5.1b). The wavenumber
92
r (m)
0.000 0.003 0.008 0.015 0.023 0.032
-2
a)
10
b)
10
0.0005
Ew2
P
c)
Ew1
-7
-3
10
w w
Pw
E(k) (m3 /s2 )
D (m2 s-3 )
0.0002
D
0.0001
D
0
P(f) (m2 s-1 )
0.0004
0.0003
0
0.02 0.04 0.06 0.08
r
2/3
2/3
(m
)
0.1
-8
2
w
1
2
10
-5
10
f-5/3
10
-5/3
10
1
-4
-6
w1
w2
vv
Puu
P
k
95% conf.
10
3
-1
k (m )
10
-2
0
10
f (Hz)
2
10
Figure 5.1: a) Structure function D for vertical velocity estimates 𝑤1(triangles) and
𝑤2 (circles). Solid lines indicate fits of equation (7). b) Wavenumber
spectrum averaged over 24-minute record. c) Temporal frequency
spectrum. Error bar indicates 95% confidence band of spectral estimate.
spectrum displays a 𝑘 −5/3 slope, indicative of the inertial subrange, and a noise floor.
A larger noise level is again observed for 𝑤2 . The noise floor was estimated as the
spectral energy level at the largest resolved wavenumber. 𝜖 was then calculated from
equation (5.1) by averaging (𝐸(𝑘) − noisefloor)𝑘 5/3 over the inertial subrange
(300 ≤ 𝑘 ≤ 1100).
The one-sided frequency spectrum was calculated for all three velocity
components at the center bin of the vertical velocity profile using Welch’s method, by
dividing the velocity records into 67 detrended segments with 50% overlap, tapered
using a Hamming window (169 degrees of freedom; Figure 5.1c). The frequency
spectra of both vertical velocity estimates are in good agreement. The inertial
subrange, with a frequency decay of 𝑓 −5⁄3 , was observed over a large range of
frequencies in the vertical velocity components, but was obscured in the horizontal
93
velocity components by the higher noise level. The turbulence dissipation rate was
calculated from the measured frequency spectrum using the frozen turbulence
hypothesis. The noise floor was calculated as the average spectral energy level for
𝑓 ≥ 47 𝐻𝑧 and 𝜖 was calculated from equation (5.4) by averaging (𝑃𝑤𝑤 (𝑓) −
noisefloor)𝑓 5/3 over the inertial subrange (4 ≤ 𝑓 ≤ 20) [similar method as
Trowbridge and Elgar, 2001 and Raubenheimer et al., 2004]. The noise floor of the
horizontal velocity components was 12 times larger than the noise floor of the vertical
components, in agreement with estimates based on the transformation matrix of the
velocimeter. The dissipation rate was also calculated for the high and low limits of the
95% confidence band of the frequency spectrum (Table 5.2).
The larger deviations in the structure function for 𝑤2 and the higher noise floor
in the wavenumber spectrum are both indications that vertical velocity measurements
𝑤2 (derived from transducers located on prongs that were aligned with the flow) had a
higher noise level than 𝑤1 (derived from transducers located on prongs that were
perpendicular to the flow). Despite the higher noise level, dissipation estimates
derived from 𝑤1 and 𝑤2 are in good agreement for all three methods. The difference
between dissipation rates estimated from the three methods is on the same order of
magnitude as the difference between dissipation rates estimated from the two
independent estimates of the vertical velocity, 𝑤1 and 𝑤2 , indicative of the instrument
accuracy (Table 5.2). Dissipation estimates based on the structure function and
wavenumber spectrum approaches also fell within the dissipation estimates based on
the 95% confidence interval of the frequency spectrum. The agreement between the
three methods in the controlled laboratory experiment indicates that the structure
94
function is a valid method to estimate the turbulence dissipation rate from the field
measurements using this velocimeter.
5.2.3
Field Experiment and Data Processing
Velocity measurements were collected during the BeST field study (see
Chapter 3 for a full description of the field site) using three Nortek Vectrino-II pulsecoherent acoustic Doppler profiling velocimeters of the same type as used in the
laboratory validation (Section 5.2.2). As in the laboratory test case, the velocimeters
collected vertical profiles of all three velocity components over a range of 0.030 m
with a resolution of 0.001 m. A coordinate system was defined with the 𝑥-axis in the
cross-shore direction with values increasing onshore, the y-axis in the alongshore
direction with values increasing to the north and the z-axis in the vertical direction
with values increasing upwards. A local vertical datum 𝑧 = 0 m was defined at the
bed level below the velocimeters at the beginning of each high-tide measurement
cycle. Flows occurred at elevations lower than z = 0 m since the bed frequently
eroded below this level over the course of a measurement cycle.
The three velocimeters were deployed at the same cross-shore location with
alongshore offsets of approximately 0.20 m (Figure 5.2). The elevation of the sensors
was chosen so that the profiling section of the three velocimeters was positioned at
nominally -0.01 m ≤ z ≤ 0.02 m, 0.015 m ≤ z ≤ 0.045 m and 0.04 m ≤ z ≤ 0.07 m
respectively at the beginning of each tide. Velocities were sampled at 100 Hz with
measurements synchronized across all velocimeters. The water level at the
velocimeter location was recorded at 4 Hz with a Druck PTX1830 pressure transducer
buried 0.05 m below the bed. The bed became continuously inundated as the water
95
Figure 5.2: Main instrument rig with three profiling velocimeters indicated by white
arrows.
level approached its maximum at high tide, meaning that both swash zone and inner
surf zone conditions are contained in the dataset.
Bed level changes of up to 0.03 m were observed at minute timescales near the
instrument array during the field study. Frequently updated estimates of the bed level
were therefore necessary to correctly situate velocity measurements with respect to the
bottom boundary. The lowest velocimeter performed a bottom scan based on the
reflected amplitude of an acoustic pulse transmitted and received by the center
transducer of the velocimeter at a sampling rate of 2 Hz (interleaved with velocity
measurements). The reflected amplitude was acquired at 1 ⋅ 10−3 m resolution for
elevations of −3 ⋅ 10−2 m ≤ 𝑧 ≤ 3 ⋅ 10−2 m. The bed eroded below the scanning
range during several tidal cycles. Dissipation estimates when the bed level was below
the scanning range were not retained in the analysis of the vertical dissipation rate
profile (section 5.3.3). Bed levels estimated using the center beam reflection showed
96
good overall agreement with bed levels estimated from the velocity profile, which
approaches zero at the non-moving bed due to the no-slip condition.
Velocity records were quality-controlled by rejecting samples with a reflected
beam amplitude below -40 dB or an instrument correlation score below 50%, or when
the water level measured by the pressure transducer was below the velocimeter probe.
Spikes were then removed from the velocity record using the phase-space despiking
method [Goring and Nikora, 2002; Wahl, 2003]. Quality control based on a
combination of a correlation score cut-off and phase-space despiking was chosen
because turbulence statistics are better preserved by a phase-space despiking method
[Mori et al., 2007]. However, the phase-space despiking method alone was not
sufficient to remove all spikes from the velocity record since it was developed for
quasi-stationary flows that are less variable and less prone to bubble contamination
than swash flows. Gaps of up to 10 samples (0.1 s) in the quality-controlled velocity
record were closed using linear interpolation [Elgar et al., 2005]. An extensive
discussion of quality-control of ADV data for calculating turbulence dissipation is
given by Feddersen et al. [2010]. The quality-control criteria proposed by Feddersen
et al. [2010] are specific to the frequency spectrum method and were therefore not
used here.
The structure function 𝐷�𝑧𝑖 , 𝑟𝑗 , 𝑡� and the fit to equation (5.7) were calculated
at time intervals of 1.25 s for each of the two independent measurements of vertical
velocity 𝑤1 and 𝑤2 . Time averaging in equations (5.5-6) was performed over a 2.5 s
window, leaving a 50% overlap in the time averaging segments. Velocity profiles
from the three velocimeters were not combined into a single profile to calculate
𝐷�𝑧𝑖 , 𝑟𝑗 , 𝑡� since velocity differences between sensors due to the alongshore offset
97
artificially increase 𝐷�𝑧𝑖 , 𝑟𝑗 , 𝑡�. Instead, fits of equation (5.7) were made for velocity
profiles from each velocimeter individually. Dissipation estimates were rejected when
the structure function was determined for less than 5 values of 𝑟𝑗 (in order to make a
robust fit of equation 5.7), if more than 20% of the velocity values in the 2.5 s time
average did not meet quality control criteria, or if the goodness-of-fit of equation (5.7),
characterized by the coefficient of determination R2, was less than 0.70. For an
elevation bin 𝑧𝑖 , the central differences technique then required at least 5 velocity time
series from elevation bins above and below 𝑧𝑖 . Therefore, 𝐷�𝑧𝑖 , 𝑟𝑗 , 𝑡� was only
calculated for the central 21 bins of each 31-bin velocity profile. As a result, 𝜖 was
estimated at elevations down to 5 ⋅ 10−3 m above the bed level and there was a
5 ⋅ 10−3 m gap between the vertical dissipation profiles from the different
velocimeters. R2-values for the structure function fit (equation 5.7) exceeded 0.70 for
89% of all dissipation estimates in the entire dataset, indicating that the 𝐷~𝑟 2/3
structure function scaling [equation (5.3)] was generally valid under field conditions.
A single time-series of the vertical-averaged dissipation rate 𝜖̅ was defined as the
average of all available 𝜖 values in the vertical dissipation profile measured by the
three velocimeters. 𝜖̅ does not equal the depth-averaged dissipation rate (defined over
the entire water column) since the velocimeters captured only the lower 0.065 m
(nominally) of the water column.
A Sony DFW-X710 IEEE1394 visible-band camera with a 1024 x 768 pixel
array was deployed on a 4 m tower on the dune ridge. During tide 5, the camera
recorded 5-minute image sequences at 5 frames per second every 15 minutes during
daylight hours, for a total of 25 minutes of video. Video collection was only
98
Figure 5.3: Example image of the video collection, including location of pixel
intensity time series points and timestack transect. Image brightness and
contrast was enhanced compared to raw image for increased visibility.
successful during tide 5 due to limited daylight hours and unfavorable weather
conditions during other tidal cycles. Images were converted to grayscale and georeferenced using surveyed ground control points [Holland et al., 1997]. Each image
sequence was geo-referenced to the vertical elevation of the mean water level during
the 5-minute sequence to account for rising water levels during the 1.25 hours when
imagery was collected. The brightness variability in the image set due to varying
cloud cover was compensated by histogram equalization of each image to a constant
reference image. Two products were derived from the image set: a pixel intensity time
series, calculated as the mean pixel intensity of 5 pixels at the cross-shore location of
the velocimeters and a cross-shore pixel intensity timestack, defined as the pixel
99
intensity on a shore-normal transect (Figure 5.3). Pixels where the instrumentation
scaffolding blocked the view of the water surface were removed from the transect.
Pixel intensities were normalized from 0 (zero intensity; black) to 1 (maximum
intensity; white).
5.3
5.3.1
Results
Time Series Excerpt
A time series excerpt of measurements collected during tide 5 is displayed in
Figure 5.4, showing an infragravity-timescale swash event with a duration of 117 s.
The bed level (black line in panels d and e) varied from 𝑧 = −0.01 m to 𝑧 =
−0.015 m. As in the entire dataset, high goodness-of-fit values were found for the
structure function fit (Figure 5.4d). Poorer agreements between the measured
structure function and the 𝐷~𝑟 2/3 scaling were mostly confined near the bed. Several
cycles of uprush (onshore-directed motion) and backwash (offshore-directed motion)
with durations between 10 s and 25 s occurred during the excerpt, superimposed on
infragravity motion that reached a maximum free surface elevation 𝜂 = 0.50 m (Figure
5.4a,b). Velocity and dissipation rate measurements were frequently interrupted upon
bore arrival (flow reversal from negative to positive velocity) due to air bubbles
entrained by the bores, potentially obscuring large turbulence dissipation rates (gaps in
Figure 5.4b-e). The dissipation rate was typically largest following bore arrival (e.g.
at t = 52 s; 85 s), then decayed rapidly (see Section 5.3.2), and increased again during
the backwash (e.g. t = 16 s; 66 s; 100 s). Measured turbulence dissipation rates in the
excerpt varied between 4 ⋅ 10−5 m2 /s 3 and 6 ⋅ 10−3 m2 /s 3 . In the entire dataset,
99% of the dissipation rate measurements fell between 6 ⋅ 10−5 m2 /s3 and 8 ⋅
100
10−3 m2 /s 3 . Little variability was observed between mean dissipation estimates from
the three velocimeters, deployed with 0.025 m vertical offsets, indicating that the
turbulence was well-mixed over this small vertical range (Figure 5.4c,e). No
Figure 5.4: Time series excerpt of measurements taken during tide 5 at 16:33:10 –
16:35:10 UTC on 12 October 2011. a) Free surface elevation 𝜂. b)
Cross-shore velocity at z = 0.02 m. c) Turbulence dissipation estimates
averaged across the vertical for lower (blue line), middle (red line) upper
(green line) velocimeter. d) Coefficient of determination 𝑅 2 of equation
(5.7) fit. e) Base-10 logarithm of turbulence dissipation 𝜖. Black line in
panels d and e indicates bed level estimate from velocimeter bed level
scan. f) Normalized pixel intensity from video camera. Thick black lines
indicate high pixel intensity events.
101
distinguishable trend was observed in the dissipation rate on infragravity timescales.
Figure 5.4f displays the normalized pixel intensity time series 𝐼 obtained from the
video imagery. Inspection of the grayscale images showed that the baseline pixel
intensity (𝐼 ≈ 0.27) in the time series corresponded with the sea surface when no
breaking occurred. High pixel intensity events, defined as instances when the pixel
intensity exceeded a threshold value of I = 0.32, were indicative of aerated wave
breaking or remnant foam (thick black lines in Figure 5.4f) [Aarninkhof and Ruessink,
2004; Haller and Catalán, 2009]. Several high pixel intensity events occurred during
the excerpt (𝑡 ≈ 3 s, 33 s, 50 s, 71 s, 83 s, 107 s), often coinciding with interruptions
in the velocity measurements (since air bubbles both caused high pixel intensities in
the video observations and disrupted the acoustic velocity measurements). The
surface signature of bores is also visible in the pixel intensity timestack corresponding
to the time series excerpt (Figure 5.5). Shoreward-propagating bores are seen as high
pixel intensity (white) diagonal streaks in the timestack. High pixel intensity events in
the pixel time series (Figure 5.4f and black transect line in Figure 5.5) were associated
with bores that propagated over tens of meters and generated a rapid increase in the
free surface elevation time series (Figure 5.4f). Another peak in the dissipation rate
time series occurred at 𝑡 = 21 𝑠. Free surface and cross-shore velocity time series
(Figure 5.4a,b) indicate that this dissipation event was associated with a propagating
wave but the surface signature of the wave is not visible in the local pixel intensity
time series (Figure 5.4f). The timestack shows that a broken wave indeed propagated
onshore, but lost most of its surface signature before it reached the location of the
velocimeters. This illustrates that broken waves may still carry significant turbulence
dissipation even after the pixel-intensity surface signature has disappeared.
102
Figure 5.5: Pixel intensity timestack. Black line indicates cross-shore position of the
velocimeters.
Despite the interruptions in the dissipation time series due to bubbles, the
relationship between in situ-measured dissipation rates and remotely-sensed pixel
intensities is still demonstrable. During the 25 minutes when video imagery was
recorded, vertically-averaged in-situ dissipation rates 𝜖̅ that occurred during a high
pixel intensity event or up to 0.5 s after a high pixel intensity event were 1.8 times
larger than dissipation rates occurring outside of high pixel intensity events, a
significant difference at the 99% level. Conversely, of the 5% highest 𝜖̅ values
measured during video collection, 57% occurred during, or up to 0.5 s after, a high
pixel intensity event.
103
5.3.2
Dissipation Decay Rates
Turbulence dissipation following swash-zone bores has been shown to decay at
a rate similar to grid turbulence [Cowen et al., 2003; Zhang and Liu, 2008]. Grid
turbulence is characterized by unidirectional flow passing through a grid obstacle that
produces a large amount of turbulence. Beyond the grid, there is no additional
turbulence production and the turbulence in the center of the flow decays freely,
without interacting with the flow boundaries. The analogous case in the swash zone is
turbulence decaying freely after being generated by a bore. Several factors may
complicate the decay behavior of bore-generated turbulence in natural conditions with
irregular wave forcing. Bores and non-breaking waves with various heights propagate
shoreward at different time intervals and may interact with each other, and bed shear
may cause additional turbulence production. Still, turbulence dissipation rates were
observed to decay rapidly after bore arrival (e.g. t = 24 s, 52 s, 85 s, 110 s in Figure
5.4).
Isolated bore events were selected to study the simplest case of the decay of
turbulence dissipation. An isolated bore event consists of a rapid water level increase
(bore arrival), followed by a gradual water level decrease (backwash) with no
interaction with secondary bores. Figure 5.6 displays a 180 s time series excerpt that
included 3 isolated bore events (Figure 5.6a). Even in the swash zone, incident-band
wave events do not necessarily start or end with a dry bed (zero water depth) since
bores propagate superimposed on infragravity motions. An isolated bore event was
identified when a rise in free surface elevation of more than 0.08 m was detected in a
timespan of two seconds or less (bore arrival). The event start time 𝑡0 was defined at
the center of the water level increase (Figure 5.6a, dots). The bore event was retained
until a new increase was observed in the free surface elevation time series, i.e. when
104
Figure 5.6: Time series excerpt of turbulence dissipation decay rates, recorded
during tide 8 at 06:28:56 – 06:31:58 UTC on 14 October 2011. a) Free
surface elevation 𝜂. Dots indicate start times 𝑡0 for three bore-dissipation
events. b) Cross-shore velocity measured at 0.02 m above the bed. c)
Measured vertically-averaged dissipation rate 𝜖̅ and fitted decay rates
according to equation (5.9). 𝑅 2 goodness-of-fit was 0.92, 0.87 and 0.94
for the three events, respectively.
𝑑ℎ
𝑑𝑡
> 0.01 m/s (indicative of a secondary bore or wave which may alter the free
turbulence dissipation decay). Bore events with a duration of less than 7 s seconds
were not utilized in the analysis so that a minimum of 5 𝜖̅ estimates were made during
the bore decay. The selection criteria for simple bore events were based only on the
water level time series so that no a priori assumptions on the velocity or turbulence
field were made. A total of 468 simple bore events were retained from the 10 tidal
cycles.
105
The decay of turbulence dissipation starts once the bore has passed the sensor
location (no more turbulence production due to the bore) and the turbulence is
vertically distributed across the water column. The depth-averaged dissipation rate 𝜖̅
was therefore analyzed for each of the isolated bore events from 2 seconds after the
start time of the event until the minimum dissipation rate was reached (end of the
decay). A power-law decay curve similar to grid turbulence decay was fit to the
vertical-averaged dissipation rate:
𝑡 𝑛
𝜖̅(𝑡) = 𝜖̅0 �𝑡 �
0
(5.9)
where 𝜖̅0 and 𝑛 are free fitting parameters. The grid-dissipation law showed excellent
agreement to the measured dissipation rate 𝜖̅ for the three bore events displayed in
Figure 5.6 (𝑅 2 = 0.92, 0.87 and 0.94, respectively). The decay exponent n for event
1, 2 and 3 was -0.78, -2.00 and -1.76 respectively. Event 2 and 3 display a clear
uprush and backwash phase in the cross-shore velocity time series (Figure 5.6b). In
contrast, the velocity during event 1 was almost exclusively negative (offshoredirected) meaning that event 1 consisted of a bore propagating shoreward against an
infragravity-timescale backwash. This explains why the value of n for event 1 was
strongly different from event 2 and 3 and from the value for grid turbulence (n = -2.3).
During event 3, the dissipation rate began to increase again toward the end of the
backwash (𝑡 ≈ 160 𝑠), which was likely due to bed-generated turbulence. The mean
𝑅 2 for all 468 events was 0.81, with 70% of all bore events exhibiting a good fit to the
decay model (𝑅 2 > 0.80). The mean decay exponent 𝑛 was -1.94 (standard deviation
0.44) for events with 𝑅 2 > 0.80.
106
5.3.3
Vertical Dissipation Rate Profiles
The vertical dissipation profile provides an indication of the dominant
turbulence production mechanisms. However, instantaneous dissipation profiles are
prone to noise and an appropriate averaging mechanism is needed to generate
characteristic turbulence dissipation profiles. Unlike laboratory studies of swash flows
generated by regular wavetrains or dambreaks, the velocity and free surface elevation
time series in the field study were irregular and strongly influenced by infragravitytimescale motions. It was therefore difficult to combine swash events for ensembleaveraging. Instead, the vertical structure of the turbulence dissipation field was
analyzed by phase space averaging (PSA) [Puleo et al., 2003; Foster et al., 2006b].
Eulerian velocity measurements of a purely sinusoidal wave follow an elliptical
trajectory in the two-dimensional phase space of cross-shore velocity and acceleration.
Binning dissipation measurements under an irregular wave train according to the
measured velocity and acceleration is analogous to wave phase-averaging. The crossshore velocity time series at the center bin of the velocity profile measured by the top
velocimeter profile (nominally at z = 0.055 m) was smoothed by convolution in time
with a Tukey window with a width of 101 samples (1.01 seconds) and a shape factor
of 0.5 in order to create a free-stream velocity time series 𝑢∞ (𝑡) and an acceleration
∞ (𝑡)
time series 𝑑𝑢𝑑𝑡
. Measured vertical profiles were grouped in velocity bins with bin
widths of 0.5 m/s and centers at -1.5 m/s; -1.0 m/s; -0.5 m/s; 0.0 m/s; 0.5 m/s and 1.0
m/s and acceleration bins with bin widths of 0.5 m/s2 and centers at -0.75 m/s2; -0.25
m/s2; 0.25 m/s2; and 0.75 m/s2 (Figure 5.7). All vertical dissipation profiles were
referenced with respect to the elevation above the instantaneous bed level z’ at the
time of the dissipation measurement, 𝑧’ = 𝑧 − 𝑧𝑏𝑒𝑑 (𝑡). Dissipation profiles were not
included when the bed level could not be determined or when no dissipation rates
107
were available for the lowest 0.01 m of the water column. Fluctuations in the bed
level 𝑧𝑏𝑒𝑑 meant that gaps between turbulence dissipation profiles from the three
velocimeters (Figure 5.4e) occurred at different elevations above the bed, so that no
gaps occurred in the combined profiles after phase-space averaging. The average
number of dissipation measurements N for each phase space-averaged vertical profile
is displayed in each panel. Points were not included in the profile when less than 50
individual dissipation estimates were combined in the average.
Figure 5.7: Phase space averaged vertical dissipation profiles (black solid line) and
25% and 75% percentiles of averaged profiles (gray dotted lines). Axes
labels and scales are identical for all panels. Downslope gravitational
∞ (𝑡)
= −0.22 m/s 2 . N-values indicate
acceleration corresponds to 𝑑𝑢𝑑𝑡
average number of dissipation rates used in the vertical profile. Panel
indicated by bold frame was compared against log-layer scaling (Section
5.4.4).
108
Swash-zone velocity time series typically display a sawtooth shape with a
roughly constant acceleration equal to the downslope gravitational acceleration. The
downslope gravitational acceleration −𝑔 sin 𝛽 was −0.22 m/s2 with the beach slope
𝛽 of 1:45 around the velocimeters and the gravitational acceleration 𝑔 = 9.81 m/s2 .
As a result, the largest number of measurements in the dataset occurred in the
∞ (𝑡)
≈ −0.25 m/s 2 (third row in Figure 5.7). Dissipation rates were
acceleration bin 𝑑𝑢𝑑𝑡
higher during the uprush than during the backwash by 6%-106% for equivalent
acceleration and velocity bins. For example, dissipation rates in the uprush bin
�𝑢∞ ≈ 0.50 m⁄s , 𝑑𝑢𝑑𝑡∞ ≈ −0.25 m/s2 �, were 70% larger than in the corresponding
backwash bin �𝑢∞ ≈ −0.50 m⁄s , 𝑑𝑢𝑑𝑡∞ ≈ −0.25 m/s2 �. Dissipation rates increased
upwards during the uprush (𝑢∞ > 0 m/s), indicative of surface-generated turbulence.
Conversely, dissipation rates during strong backwash conditions (𝑢∞ ≤ −1 m/s)
decreased upwards, indicative of bed-generated turbulence. The transition from
surface-dominated dissipation to bed-dominated dissipation occurred when 𝑢 ≈
−0.5 m/s, indicating that the early stages of backwash were still dominated by surface
processes. It is noted that only near-bed (0 m ≤ 𝑧 ′ ≤ 0.06 m) dissipation rates were
measured. Dissipation rates higher in the water column, and particularly above the
wave trough level, may have been larger than the near-bed dissipation rates. For this
reason, no attempt was made to compare the near-surface turbulence dissipation rate
profile to surf-zone scalings such as proposed by Feddersen et al. [2012b].
5.3.4
Wave Height and Water Depth Dependence
Wave height and the (tidally modulated) local water depth are two potentially
influential factors to the average turbulence dissipation rate. Offshore significant
wave heights decreased from 2.46 m to 0.56 m over the course of the study
109
Figure 5.8: a) Turbulence dissipation as a function of local water depth. b)
Inundation percentage. Error bars show variability (1 standard deviation)
between results from 10 tide measurement cycles.
(Table 3.1). However, the turbulence dissipation rate, averaged over both the vertical
profile and in time over each high tide measurement cycle was not significantly
correlated with the mean offshore wave height (Pearson correlation coefficient r =
0.15). Similarly, no significant correlation was found between the tidally-averaged
2
turbulence dissipation rate and the offshore wave energy, represented by 𝐻𝑟𝑚𝑠
(r =
0.05). The study site was a dissipative beach where wave breaking occurred over
several wavelengths, indicating a saturated surf zone (Figure 5.3). Wave energy in the
gravity-wave band (wave period less than 20 s) in a saturated surf zone is depthlimited and roughly independent of offshore wave forcing [Thornton and Guza, 1982].
The local water depth ℎ�, defined as the 30-minute running average of the water
depth recorded by the pressure transducer, varied over the course of the rising and
110
falling tide and peaked at 0.23-0.42 m during the different tides. Vertically-averaged
turbulence dissipation values were binned according to depth with bin widths of 0.05
m. Measurements for the two smallest depth bins, ℎ� < 0.1 m, were combined because
of the small number of valid measurements. Turbulence dissipation rates (Figure 5.8)
remained roughly constant in the inner surf zone (ℎ� > 0.20 m) and increased with
decreasing water depth in the swash zone (ℎ� < 0.20 m). However, there was
considerable variability among the 10 tidal cycles as demonstrated by the error bars (1
standard deviation). Distributions of vertically-averaged dissipation rates for the
different depth bins show that dissipation rates followed a lognormal distribution for
all depths with a similar shape among the different tidal cycles (Figure 5.9). Within
each depth bin, 𝜖̅ varied by approximately 2 orders of magnitude, consistent with
earlier studies (Table 5.1).
Figure 5.9: Distributions of turbulence dissipation rates binned according to local
water depth. Different curves display results from the 10 tide
measurement cycles.
111
5.4
5.4.1
Discussion
Analysis Methods
Turbulence dissipation rates calculated from the structure function, the
wavenumber spectrum, and the frequency spectrum showed excellent agreement in an
idealized laboratory test case. The structure function was chosen to analyze the field
data for several reasons. Estimating the dissipation rate based on the frequency
spectrum invokes the frozen turbulence hypothesis, which has a number of
disadvantages (see Section 5.1). In the laboratory test case, the wavenumber spectrum
displayed the inertial subrange for only 4 points and over less than one order of
magnitude of 𝐸(𝑘) before flattening to a noise floor (Figure 5.1). In contrast, the
structure function followed the inertial subrange over 11 data points, meaning that a
more robust fit can be performed to estimate 𝜖. Furthermore, the structure function
𝐷�𝑧𝑖 , 𝑟𝑗 , 𝑡� can be calculated at different elevations 𝑧𝑖 to estimate the vertical
dissipation profile, while the wavenumber spectrum cannot be calculated at different
elevations unless the number of points used to calculated the spectrum is halved to 16
(leaving only 8 points in the one-sided wavenumber spectrum). Lastly, calculating
either the wavenumber or the frequency spectrum introduces spectral leakage, whereas
the structure function is calculated directly from the measured vertical velocity profile.
The time averaging window of 2.5 s, used in equations (5.5-6) to calculate D,
was chosen as a compromise between achieving a robust estimate of the timeaveraged quantities and maintaining sufficient resolution to resolve the temporal
evolution of 𝜖 (e.g. the rapid decay of bore-generated turbulence). The sensitivity to
the choice of time averaging window length was examined by repeating the analysis
with a larger (3.5 s) and smaller (1.5 s) averaging window for tide 5 (Figure 5.10).
112
Figure 5.10: Repeat of time series excerpt displayed in Figure 5.4. Time series
excerpt of measurements taken during tide 5 at 16:33:10 – 16:35:10 UTC
on 12 October 2011. a) Free surface elevation 𝜂. b) Cross-shore velocity
at z = 0.02 m. c-e) Turbulence dissipation estimates calculated using an
averaging window of 2.5 s (c), 1.5 s (d) and 3.5 s (e) averaged across the
vertical for lower (blue line), middle (red line) upper (green line)
velocimeter.
Calculating D and 𝜖 with a 3.5 s averaging window led to reduced noise in the 𝜖
timeseries but certain features in the timeseries were smoothed out due to the reduced
temporal resolution. Conversely, reducing the averaging window to 1.5 s improved
the temporal resolution but increased the noise in the dissipation rate time series. The
113
overall effect of varying the averaging window between 1.5 s and 3.5 s was small and
general trends in the 𝜖 timeseries were identical, meaning that dissipation rate
estimates were not overly sensitive to the choice of the averaging window length.
Mean vertical velocities in the swash zone are small so the effect of subtracting the
mean velocity (equation 5.6) is likely to be small as well [Aagaard and Hughes,
2006].
The 𝐷 ∝ 𝜖 2⁄3 𝑟 2⁄3 scaling for the structure function (equation 5.3) is only valid
in the inertial subrange, defined by 𝑙𝑘 ≪ 𝑟 ≪ 𝑙𝑒 where 𝑙𝑘 is the Kolmogorov length
scale and 𝑙𝑒 is the length scale of the largest eddies. The separation distance r used to
calculate 𝐷 (equation 5.6) ranged between 2 ⋅ 10−3 m and 3 ⋅ 10−2 m and dissipation
rates between 6 ⋅ 10−5 m2 /s 3 and 8 ⋅ 10−3 m2 /s 3 were found. The Kolmogorov
4
𝜈3
length scale, defined as 𝑙𝑘 = � [Pope, 2000] (with 𝜈 = 1.15 ⋅ 10−6 m2 /s as the
𝜖
kinematic viscosity of water at 15 °C), ranged between 1.18 ⋅ 10−4 m and 4.0 ⋅
10−4 m, an order of magnitude smaller than 𝑟. The length scale of the largest eddies
has been estimated in the swash zone as the distance between the measurement
location and the bed [Flick and George, 1990] and as 0.4 times the local water depth
[Sou et al., 2010]. The dissipation rate was measured at elevations from 0.005 m up
to roughly 0.065 m above the bed and when local water depths were between 0.07 m
and 0.85 m. The separation distance 𝑟 was therefore always smaller than the largest
eddy length scale 𝑙𝑒 . As a result, the calculated structure function fell within the
inertial subrange meaning that the application of equation (5.3) was valid, as is
demonstrated by the high R2 values (Figure 5.4). The separation distances for 𝜖
estimates near the bed were of the same size as the largest eddy length scale, which
may explain the occasionally lower 𝑅 2 values near the bed.
114
5.4.2
Dissipation Rate Magnitude
Observed instantaneous dissipation rates ranged between 6 ⋅ 10−5 m2 /s3 and
8 ⋅ 10−3 m2 /s 3 with a mean dissipation rate of 1.2 ⋅ 10−3 m2 /s3 to 1.3 ⋅ 10−3 m2 /s 3
depending on local water depth. Turbulence dissipation rates were measured in the
swash zone and in the inner surf zone in both the BeST field study and in the field
study by Raubenheimer et al. [2004]. Dissipation rates observed during the BeST
field study were an order of magnitude smaller than swash-zone dissipation rates
reported by Raubenheimer et al. [2004]. However, inner surf zone dissipation rates
observed in the BeST study are of same order of magnitude as rates observed
throughout the surf zone in other studies, whereas those observed by Raubenheimer et
al. [2004] are higher than most previously reported values (Table 5.1). The
differences in observed dissipation values in different studies are likely due to
differences in the field site morphologies (e.g. beach slope), offshore wave forcing,
sensor position in the water column and due to the different methods used to estimate
𝜖.
5.4.3
Vertical Dissipation Profile
The dissipation rate increased upwards during the uprush, indicating that bore-
generated turbulence was likely the dominant mechanism, and decreased upwards
during backwash, indicating that bed-generated turbulence was dominant. In a
laboratory study, Sou et al. [2010] observed that dissipation profiles decreased
upwards during both the uprush and the backwash in the swash zone, but the near-bed
dissipation profile increased upwards during the uprush in the inner surf zone, near the
boundary with the swash zone. Other laboratory studies of swash zone turbulence
displayed only vertical profiles of turbulent kinetic energy (TKE), not 𝜖. Although
115
TKE and 𝜖 are not equivalent, these profiles still provide a useful indication of the
dominant processes. Petti and Longo [2001] found a similar structure for the vertical
TKE profile in a laboratory study as observed for 𝜖 in this chapter. O’Donoghue et al.
[2010] found that TKE was nearly depth-uniform on a smooth beach slope, but that it
decreased upwards during both the uprush and the backwash for a rough beach with a
much larger grain size (𝑑 ≈ 5 ⋅ 10−3 m) than the study site discussed here (𝑑50 =
0.33 ⋅ 10−3 m), highlighting the importance of bed roughness in the balance between
bed-generated and surface-generated turbulence. In summary, vertical turbulence
dissipation profiles observed in this study are generally consistent with previously
observed profiles.
5.4.4
Turbulence Damping by Density Stratification
If turbulence dissipation was dominated by bed-generated turbulence during
the backwash, the vertical dissipation profile may have followed the “log-layer”
scaling [Tennekes and Lumley, 1972]. The log-layer scaling assumes that turbulence
production 𝒫 is balanced by dissipation:
𝑢3
∗
𝜖 = 𝒫 = 𝜅𝑧′
(5.10)
where 𝑢∗ = �𝜌𝜏𝑏 is the friction velocity, 𝜏𝑏 is the bed shear stress, 𝜌𝑓 = 1025 kg/m3
𝑓
is the fluid density and 𝜅 = 0.41 is the von Kármán constant. Dissipation rates in the
surf zone were found to be greater than the log-layer prediction due to breaking-wave
generated turbulence, even under strong alongshore flows [Feddersen et al., 2007;
Grasso et al., 2012]. The log-layer dissipation scaling was examined for the phasespace-averaged backwash bin −1.25 m/s ≤ 𝑢∞ ≤ −0.75 m/s, −0.50 m/s2 ≤
116
𝑑𝑢∞
𝑑𝑡
≤ 0 m/s 2 (panel indicated by bold frame in Figure 5.7). The friction velocity
was estimated using a quadratic drag law:
𝜏𝑏
𝑢∗ = �𝜌
𝑓
1
= �2
2
𝜌𝑓 𝑓𝑢∞
𝜌𝑓
(5.11)
The backwash friction factor f was estimated for this field dataset from log-law
velocity profiles as 𝑓 = 0.021 [Puleo et al., 2014b], yielding 𝑢∗ = 0.1 m/s. Based on
this friction velocity estimate, the log-layer turbulence dissipation estimate [equation
5.10)] for 𝑧′ = 0.04 m is 𝜖 = 6.6 ⋅ 10−2 m2 /s 3 , two orders of magnitude larger than
the measured dissipation rate of 8.1 ⋅ 10−4 m2 /s 3 (Figure 5.7).
The large discrepancy between measured dissipation rates and rates predicted
by the log-law scaling may be partially due to uncertainties in the friction factor, the
elevation above the bed z’ of the velocity measurements, because the flow acceleration
𝑑𝑢∞
𝑑𝑡
was not zero or because the turbulence may have still been developing
(production larger than dissipation) throughout the backwash. Another factor in the
balance between turbulence production and dissipation is sediment-induced density
stratification [Ross and Mehta, 1989; Winterwerp, 2001; Hsu and Liu, 2004]. The
swash zone is characterized by some of the highest sediment concentrations found in
the coastal environment and vertical sediment concentration gradients near the bed of
𝑘𝑔
−10 3
O( 0.01𝑚𝑚
= −1000 kg/m4 ) have been observed [Puleo et al., 2000; Aagaard and
Hughes, 2006]. Suspended sediment concentrations were measured at different
elevations during this field study but the quality of the measurements was not
sufficiently consistent to obtain a detailed spatio-temporal overview of suspended
sediment concentration throughout the 10 tidal cycles. However, a detailed
117
investigation of suspended sediment concentrations on the same beach as the BeST
study [Masselink et al., 2005] showed a mean concentration profile 𝑐(𝑧′) of
𝑧′
𝑐(𝑧′) = 71 kg⁄m3 ⋅ exp �− 0.023m�
𝑑𝑐
during the backwash, yielding 𝑑𝑧 = −5.4 ⋅ 102 kg⁄m4 and 𝑑𝜌
=
𝑑𝑧
(5.12)
𝜌𝑠 −𝜌𝑓 𝑑𝑐
=−3.3⋅102 kg⁄m4
𝜌𝑠 𝑑𝑧
at
𝑧′ = 0.04 m, with 𝜌𝑠 = 2650 kg/m3 the sediment density and 𝜌 the density of the
fluid-sediment mixture. A preliminary analysis of vertical suspended sediment
concentration gradients observed during the BeST field study displayed similar
magnitudes [Puleo et al., 2014b]. An order-of magnitude estimate of the turbulence
damping due to density stratification S was calculated as
𝜈
𝑑𝜌
𝑆 = 𝜌𝜎𝑡
𝑑𝑧
𝑐
𝑔
(5.13)
where 𝜈𝑡 is the eddy viscosity and 𝜎𝑐 = 0.7 is the Prandtl-Schmidt number
[Winterwerp, 2001; Hsu and Liu, 2004]. The eddy viscosity was calculated consistent
𝑑𝜌
with the log-law scaling as 𝜈𝑡 = 𝜅𝑧𝑢∗ . With 𝑑𝑧 estimated using the result by
Masselink et al. [2005], the stratification term 𝑆 was −7.5 ⋅ 10−3 𝑚2 /𝑠 3, an order of
magnitude larger than the measured turbulence dissipation 𝜖. The importance of
density stratification was also assessed using the gradient Richardson number [Turner,
1979]:
𝑅𝑖 =
1 𝜕𝜌
𝑔
𝜌 𝜕𝑧
𝜕𝑢 2
�
𝜕𝑧
�
.
(5.14)
The phase-space-averaged velocity profile 𝑢(𝑧) is displayed in Figure 5.11. The
vertical velocity gradient
𝑑𝑢
𝑑𝑧
at 0.04 m was estimated at −3.2 𝑠 −1 . The Richardson
number was then 𝑅𝑖 = 0.38 > 0.25, meaning that turbulence was damped by density
stratification [Miles, 1961; Geyer and Smith, 1987; Trowbridge and Kineke, 1994]. It
118
0.06
z ( m)
0.04
0.02
0
-1.5
-0.5
-1
0
u (m/ s)
Figure 5.11: Phase-space averaged velocity profile for bin −1.25 m/s ≤ 𝑢∞ ≤
𝑑𝑢
−0.75 m/s, −0.50 m/s2 ≤ 𝑑𝑡∞ ≤ 0 m/s2 . Gray dotted lines indicate
25% and 75% percentiles of averaged velocity profiles.
is noted that stratification may reduce the value of κ in equation (5.10) [Winterwerp,
2001] but this does not lead to order-of-magnitude changes in the scaling analysis.
Suspended sediment concentrations have been observed to be even larger
during the uprush than during the backwash [Masselink et al., 2005] so density
stratification effects are expected to occur during both the uprush and backwash
phases of the swash cycle. Understanding the exact spatial and temporal importance
of density stratification requires a detailed, simultaneous analysis of the suspended
sediment concentrations and the turbulent kinetic energy budget that is beyond the
scope of this chapter. However, this order-of-magnitude scaling analysis demonstrates
that density stratification plays a significant role in the near-bed TKE budget in the
swash zone. Laboratory and numerical studies of swash zone turbulence that do not
include suspended sediment may therefore overestimate near-bed turbulence levels
compared to natural conditions.
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5.5
Conclusions
The turbulence dissipation rate 𝜖 in the swash zone and the inner surf zone of a
dissipative beach was investigated using a structure function method. A laboratory
test under stationary unidirectional flow displayed excellent agreement between
turbulence dissipation estimates based on the structure function, the wavenumber
spectrum and the frequency spectrum. The main advantages of the structure function
method are that it does not invoke Taylor’s hypothesis and that it provides the
opportunity to investigate the spatial and temporal evolution of the dissipation rate.
Swash zone dissipation rates were estimated between 6 ⋅ 10−5 m2 /s3 and 8 ⋅
10−3 m2 /s 3 , with a mean of 1.2 ⋅ 10−3 m2 /s3 to 1.3 ⋅ 10−3 m2 /s 3 depending on
local water depth. Dissipation rates increased slightly with decreasing water depth in
the swash zone and showed no significant correlation with offshore wave height,
consistent with a saturated surf zone. The temporal variability of the turbulence
dissipation was mainly driven by bores and a good agreement was observed between
dissipation rates and remotely-sensed pixel intensities, which were used as a proxy for
wave breaking. However, bores may carry significant turbulence dissipation even
after the breaking wave surface signature has disappeared. Dissipation rates decayed
rapidly after bore arrival following a decay rate similar to grid turbulence, in the form
of a power law with exponent -1.94.
Vertical dissipation profiles showed that turbulence dissipation was dominated
by bore-generated turbulence during the uprush and bed-generated turbulence during
the later stages of the backwash, with the transition from bore-dominated to beddominated occurring during early stages of the backwash when the cross-shore
velocity was approximately -0.5 m/s. Backwash turbulence dissipation rates,
however, were two orders of magnitude smaller than values obtained from a log-layer
120
prediction. A scaling analysis revealed that due to the large suspended sediment
concentrations in the swash zone, sediment-induced density stratification was an order
of magnitude larger than measured turbulence dissipation. This finding has important
implications for numerical and laboratory studies of swash zone turbulence, which
may overestimate near-bed turbulence levels if the effect of sediment-induced density
stratification on the turbulent kinetic energy budget is not accounted for.
121
Chapter 6
RECOMMENDATIONS FOR FURTHER RESEARCH
6.1
The Conductivity Concentration Profiler
The development of the Conductivity Concentration Profiler (Chapter 2), the
first instrument capable of measuring sediment concentration profiles in the sheet flow
layer under field conditions, opens a host of new research possibilities. First, more
datasets are to be collected of sheet flow in different environments and conditions,
including recent efforts in the Netherlands and Mexico. A few months after the first
CCP measurements were taken during the BeST field experiment (Chapter 3 and 4),
CCPs were again deployed during the Bardex II (Barrier Dynamics Experiment)
study. The aim of this experiment was to “collect a near proto-type data set of
energetic waves acting on a sandy beach/barrier system to improve our quantitative
understanding and modelling capability of shallow water sediment transport processes
in the inner surf, swash and overwash zone.” [Masselink et al., in preparation]. To this
end, a sand barrier island was constructed in the large-scale Delta Flume facility in the
Netherlands and exposed to a range of wave and water level conditions. CCPs were
deployed in the swash zone that had a steep (roughly 1:10) slope and experienced
wind-wave period (T = 8 - 12 s) dominated swash events, in contrast with the
infragravity-timescale swash events on a low-sloping (roughly 1:45) beach during the
BeST field study. Bardex II thus provided a second CCP dataset that was highly
complementary to the BeST dataset. CCPs have also been deployed on a sea-breeze
122
dominated beach in Yucatan, Mexico in March - April 2014. A deployment in the surf
zone as part of a large-scale laboratory study has been planned as well.
The CCP may also be useful for investigating other processes besides sheet
flow. The sediment concentration profiles can be used to generate continuous time
series of bed level elevation regardless of whether the sensor is submerged or
emerged, in contrast with other techniques such as acoustic or lidar measurements
which only give a bed level elevation estimate when the bed is emerged in between
swash events. Analysis of these bed level elevation timeseries [Puleo et al., 2014a]
revealed interesting fluctuations at time scales from seconds to hours. Deploying two
or three CCPs at small [O(1 m)] cross-shore separation distances would make it
possible to study fluctuations of the local bed level slope. The CCP could also be
modified to serve as a compact, field-capable conductivity profiler to measure other
parameters such as the soil moisture content (saturation level) of the beach sand
volume or to detect fresh or salt water lenses. A preliminary test has shown that the
CCP may also be capable of measuring time-varying compaction of cohesive sediment
beds. Since the conductivity probe is detachable from the main measurement board, a
new conductivity probe with a different geometry could be tailor-made for each of
these applications and remain compatible with the existing measurement board.
6.2
Sheet Flow and Swash-zone Morphology
Analysis of sediment concentration measurements in this work focused on
sheet flow during quasi-steady backwash, which is the simplest case of sheet flow
observed in the swash zone (Chapter 4). The more general case of sheet flow under
arbitrary forcing conditions (in other words, sheet flow as it occurs throughout the
entire swash cycle) is much more complicated due to additional sediment mobilization
123
mechanisms such as pressure gradients and bore turbulence and due to unsteady
forcing. Near-bed sediment mobilization by pressure gradients is known as plug flow
[Sleath, 1999; Foster et al., 2006a] and sediment transport during the uprush in the
swash zone may be a combination of sheet flow and plug flow. The Bardex II CCP
dataset, which had shorter incident wave periods, no infragravity-dominated wave
motion, and a higher CCP sampling frequency (8 Hz, compared to 4 Hz during BeST),
provided an excellent starting point for the analysis of time-varying sheet flow.
The CCP has provided measurements of the sheet flow sediment concentration
profile under field conditions with unprecedented detail. In order to calculate
sediment fluxes, however, a velocity profile in the sheet flow layer is also necessary.
The velocity profile in the sheet flow layer has been measured in laboratory
experiments, e.g., using image analysis through the side-wall of a flow tunnel [Ahmed
and Sato, 2001; Capart et al., 2002], using an in-situ boroscope [Cowen et al., 2010]
or by computing the cross-correlation of sediment concentration time series measured
at different locations along the flow [McLean et al., 2001]. Acoustic Doppler
velocimetry techniques have been optimized to penetrate into the sheet layer, but it is
uncertain if measurements can be made at high concentrations in the lower sections of
the sheet layer for natural sand particles [Hurther et al., 2011; Chassagneux and
Hurther, 2014]. None of these techniques are readily deployable in the field, either
because the measurement instruments cannot be deployed without generating
significant flow disturbance and scour, or because repeatable, controllable conditions
are required to obtain reliable velocity estimates. Puleo et al. [submitted, 2014b]
therefore calculated sediment fluxes from CCP concentration measurements and an
124
estimated velocity profile by extrapolating velocities measured above the sheet flow
layer to the non-moving sediment bed using an assumed velocity profile function.
Developing field-capable measurement techniques for velocity in the sheet
flow layer is therefore a crucial next step to improve estimates of the sediment
transport budget in the swash zone. Of the aforementioned laboratory techniques,
imaging techniques may be the most promising candidate to be adapted to field scale.
A sideways-looking high-speed (90 frames per second) video camera was deployed in
the swash zone during the Bardex II study as a pilot test. A plexiglass side-wall was
placed parallel to the cross-shore flow so that the flow disturbance created by the
camera housing did not influence the flow field in the camera field of view. The highspeed imagery will be analyzed using Particle Image Velocimetry (PIV) or other
techniques to provide velocity estimates within the sheet flow layer. Cross-correlation
of two sediment concentration signals measured at different cross-shore locations
[McLean et al., 2001; Dohmen-Janssen and Hanes, 2002; Hassan and Ribberink,
2005] may also appear promising for field-scale application. However, the technique
has so far only proven successful when ensemble-averaging across several repeatable
waves was used to suppress noise and find the cross-correlation peak. Ensemble
averaging is difficult under irregular wave forcing and it is therefore uncertain whether
a cross-correlation technique can be successful in field settings. In addition,
concentration measurements must be collected at high sampling rates (1000 Hz) to
perform cross-correlation. These sampling rates are achievable using a single-point
concentration sensor such as the CCM. The CCP, which measures a 29-point
concentration profile by cycling through an array of electrodes, presently has a
125
maximum sampling rate of 8 Hz and would have to be modified significantly to
achieve high enough sampling rates to perform cross-correlation.
Measurements of sheet flow using the CCP and a yet-to-be-developed velocity
measurement solution, as well as measurements of turbulence dissipation (Chapter 5),
certainly lead to an improved understanding of the swash-zone sediment budget at the
small scale (timescales on the order of seconds and minutes). However, it is important
to bear in mind that even with improved predictions of small-scale sediment transport,
accurate predictions of sediment transport at the meso-scale (time scales of hours, days
or weeks), which are ultimately important for practical applications, are still far from
achieved. An alternative approach for developing improved models of swash zone
morphodynamics at the meso-scale may be to focus on meso-scale behavior directly
rather than as the aggregate of small scale processes. For example, Masselink et al.
[2009] suggested developing a sediment transport model based on the equilibrium
gradient in the swash zone. Measurements of local beach slope fluctuations using the
CCP (Section 6.1) would provide useful data to develop such a model.
6.3
Turbulence Dissipation Rates
Chapter 5 described how swash zone turbulence dissipation rates were
estimated from high-resolution (1 mm), small-range (3 cm) velocity profile
measurements using the structure function. Before this work, the structure function
had only been applied to large-range [O(10 m)] velocity profiles measured by
Acoustic Doppler Current Profilers (ADCPs) [Wiles et al., 2006; Mohrholz et al.,
2008; Whipple and Luettich, 2009]. The advent of a next-generation profiling acoustic
velocimeter, the Nortek Vectrino II [Craig et al., 2011], made it possible for the first
time to calculate the structure function at a smaller scale, e.g. in the shallow flows of
126
the swash zone. As these next-generation profiling velocimeters become more
widespread, the structure function method will remain a valuable tool to estimate
turbulence dissipation rates in different environments. For example, Brinkkemper et
al. [in preparation] analyzed turbulence throughout the surf and swash zones during
the Bardex II experiment and used the structure function for swash-zone dissipation
rate estimates. Similarly, Pieterse et al. [2013] estimated turbulence dissipation rates
in shallow flows over a tidal saltmarsh.
6.4
Negative Results
Recommendations for further research also come in the form of negative or
unsuccessful research results, serving as both a warning to those who consider going
down the same path and as an outtakes reel at the end of the story. During the
development of the CCP, for example, some effort was initially made to extend the
two-electrode conductivity measurement technique of Puleo et al. [2010] to operate in
both fresh and salt water. This effort proved unsuccessful, and the approach was
changed to the four-electrode conductivity measurement technique that is used in the
current version of the CCP and the CCM.
A technique was also developed to estimate cross-shore velocities throughout
the entire swash event and address the problem of backwash flows that are too shallow
for in-situ current meters. The swash-zone free surface was detected from video
images recorded by a camera looking through the side wall of a small wave flume, and
a time series of depth-averaged velocities was calculated using the volume continuity
technique validated by Blenkinsopp et al. [2010b]. While providing accurate results
[Lanckriet and Puleo, 2010], the technique was cumbersome and could only be used
in flumes with glass sidewalls. Acoustic distance meters and lidar are more
127
advantageous options than video imagery for obtaining measurements of the free
surface since they are easier to set up and can be deployed in a wider range of
environments, including large-scale laboratory and field experiments. These two
techniques have therefore become increasingly popular [Blenkinsopp et al., 2010a;
Puleo et al., 2014b].
128
Chapter 7
CONCLUSIONS
This work described field measurements and innovative measurement
techniques to improve understanding of near-bed hydrodynamics and sediment
transport in the swash zone. A first main accomplishment was the development of the
Conductivity Concentration Profiler, a new sensor that measures sediment
concentrations in the sheet flow layer. The CCP cycles through an array of electrodes
to obtain a vertical profile of electrical conductivity with a 29 mm range and a 1 mm
resolution. Laboratory measurements of natural sand suspended neutrally in a Lithium
Metatungstate (LMT) solution demonstrated that electrical conductivity of the
sediment-water mixture is a good measurement proxy for sediment concentration.
The relationship between conductivity and concentration was described accurately by
Archie’s law. A finite differences model of the current field around the CCP
conductivity probe was set up to analyze the CCP measurement volume and its
smoothing effect on sharp gradients in the concentration profile. Model results
showed that the measurement volume is large enough to contain at least several
thousand sand grains leading to a robust estimate of the sediment volume fraction, and
that sheet flow layers with a thickness of over 5 mm are accurately resolved by the
CCP. The numerical model was also used to develop a correction formula to correct
for any residual smoothing effect at larger sheet thicknesses.
129
A comprehensive field dataset of swash zone processes, including CCP
measurements in the sheet flow layer, was obtained during the BeST (Beach Sand
Transport) experiment conducted in Perranporth, UK, in October 2011.
Analysis of sheet flow measurements in this work focused on quasi-steady
backwash, which generated sheet flow that was unaffected by surface-generated
turbulence, phase lags and pressure gradients and thus was highly similar to stationary,
unidirectional sheet flow. Sheet flow sediment concentration profiles displayed a selfsimilar shape for sheet flow layer thicknesses ranging from 6 to 18 mm, with a linear
shape in the lower section of the profile and a power-law shape in the upper section.
A single curve proposed by O’Donoghue and Wright [2004a] described the profile
shape for all observed sheet flow layer thicknesses. Sheet flow layer thicknesses and
sheet load showed good correlation (r2 = 0.60 and 0.53, respectively) with the mobility
number, a measure of hydrodynamic forcing that is proportional to the flow velocity
squared. A previous analytical model to predict sheet flow layer thickness by Wilson
[1987] assumed a linear shape of the concentration profile. Incorporating the curve by
O’Donoghue and Wright [2004a] into this model provided sheet flow layer thickness
predictions that are in better agreement with previous experiments.
High-resolution near-bed velocity profiles were also measured during the
BeST field study using a new profiling acoustic Doppler velocimeter, and were used
to estimate the turbulence dissipation rate based on the structure function. The
structure function method was first validated in a laboratory test under stationary flow
and showed excellent agreement with turbulence dissipation rate estimates derived
from both the wavenumber spectrum and the frequency spectrum. Dissipation rates
observed in the swash zone ranged between 6 ⋅ 10-5 m2/s3 and 8 ⋅ 10-3 m2/s3, with a
130
mean of 1.2 ⋅ 10-3 m2/s3 to 1.3 ⋅ 10-3 m2/s3 depending on local water depth.
Dissipation rates tended to be highest immediately following bore arrival, then
decreased rapidly during the remainder of the uprush, and then increased again during
the backwash. The rapid decay of the dissipation rate following bore arrival was
similar to grid turbulence decay, and can be described by a power law with exponent
-1.94 (compared to -2.3 for grid turbulence). Large dissipation rates following bore
arrival also corresponded with high pixel intensities in remotely sensed video imagery
of the measurement site, indicative of wave breaking. The dominant turbulence
generation mechanism was investigated based on vertical profiles of the dissipation
rate. Bore-generated turbulence was the dominant source during the uprush and bed
shear was dominant during the backwash. A scaling analysis showed that sedimentinduced density stratification was an order of magnitude larger than turbulence
dissipation rates measured during the backwash, indicating that even on sandy
beaches, stratification effects are an important component in the near-bed turbulent
kinetic energy budget.
Going forward, both the CCP and the structure function method will be
important tools to quantify sheet flow sediment transport and turbulence dissipation
rates, respectively, and will be used in future field and laboratory studies.
Determining velocities in the sheet flow layer remains an important next step to
accurately determine the sediment budget in the swash zone.
131
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Appendix
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Reproduced and modified, with permission, from Lanckriet, T., J. A. Puleo, and N.
Waite (2013), A Conductivity Concentration Profiler for Sheet Flow Sediment
Transport, IEEE Journal of Oceanic Engineering, 38(1), 55 –70,
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The abstract (in part), section 2.3 (in part), section 3.2 (in part), Chapter 4,
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42, doi:10.1061/(ASCE)WW.1943-5460.0000209.
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from Puleo, J. A. et al. (2014), Comprehensive Field Study of Swash-Zone Processes.
I: Experimental Design with Examples of Hydrodynamic and Sediment Transport
Measurements, Journal of Waterway, Port, Coastal, and Ocean Engineering, 140(1),
14–28, doi:10.1061/(ASCE)WW.1943-5460.0000210.
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permission from John Wiley and Sons, from Lanckriet, T., and J. A. Puleo (2013),
Near-bed turbulence dissipation measurements in the inner surf and swash zone,
Journal of Geophysical Research: Oceans, 118(12), 6634–6647,
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